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Published: 10 December 2022

Last updated: 29 January 2023

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1

Fleming-Vázquez, Nicholas. "Functional Correlation Bounds and Optimal Iterated Moment Bounds for Slowly-Mixing Nonuniformly Hyperbolic Maps." Communications in Mathematical Physics 391, no. 1 (February 2, 2022): 173–98. http://dx.doi.org/10.1007/s00220-022-04325-w.

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AbstractConsider a nonuniformly hyperbolic map $$ T:M\rightarrow M $$ T : M → M modelled by a Young tower with tails of the form $$ O(n^{-\beta }) $$ O ( n - β ) , $$ \beta >2 $$ β > 2 . We prove optimal moment bounds for Birkhoff sums $$ \sum _{i=0}^{n-1}v\circ T^i $$ ∑ i = 0 n - 1 v ∘ T i and iterated sums $$ \sum _{0\le i<j<n}v\circ T^i\, w\circ T^j $$ ∑ 0 ≤ i < j < n v ∘ T i w ∘ T j , where $$ v,w:M\rightarrow {{\mathbb {R}}} $$ v , w : M → R are (dynamically) Hölder observables. Previously iterated moment bounds were only known for $$ \beta >5$$ β > 5 . Our method of proof is as follows; (i) prove that $$ T$$ T satisfies an abstract functional correlation bound, (ii) use a weak dependence argument to show that the functional correlation bound implies moment estimates. Such iterated moment bounds arise when using rough path theory to prove deterministic homogenisation results. Indeed, by a recent result of Chevyrev, Friz, Korepanov, Melbourne & Zhang we have convergence to an Itô diffusion for fast-slow systems of the form $$\begin{aligned} x^{(n)}_{k+1}=x_k^{(n)}+n^{-1}a(x_k^{(n)},y_k)+n^{-1/2}b(x_k^{(n)},y_k) , \quad y_{k+1}=Ty_k \end{aligned}$$ x k + 1 ( n ) = x k ( n ) + n - 1 a ( x k ( n ) , y k ) + n - 1 / 2 b ( x k ( n ) , y k ) , y k + 1 = T y k in the optimal range $$ \beta >2$$ β > 2 .

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2

Weigel, A. Gambini, and T. S. Weigel. "On the orders of primitive linear P'-groups." Bulletin of the Australian Mathematical Society 48, no. 3 (December 1993): 495–521. http://dx.doi.org/10.1017/s0004972700015951.

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A group G ≤ GLK(V) is called K-primitive if there exists no non-trivial decomposition of V into a sum of K-spaces which is stabilised by G. We show that if V is a finite vector space and G a K-primitive subgroup of GLK(V) whose order is coprime to |V|, we can bound the order of G by |V|log2(|V|) apart from one exception. Later we use this result to obtain some lower bounds on the number of p–singular elements in terms of the group order and the minimal representation degree.

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Hernández Mira, Frank A., Ernesto Parra Inza, José M. Sigarreta Almira, and Nodari Vakhania. "Properties of the Global Total k-Domination Number." Mathematics 9, no. 5 (February 26, 2021): 480. http://dx.doi.org/10.3390/math9050480.

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A nonempty subset D⊂V of vertices of a graph G=(V,E) is a dominating set if every vertex of this graph is adjacent to at least one vertex from this set except the vertices which belong to this set itself. D⊆V is a total k-dominating set if there are at least k vertices in set D adjacent to every vertex v∈V, and it is a global total k-dominating set if D is a total k-dominating set of both G and G¯. The global total k-domination number of G, denoted by γktg(G), is the minimum cardinality of a global total k-dominating set of G, GTkD-set. Here we derive upper and lower bounds of γktg(G), and develop a method that generates a GTkD-set from a GT(k−1)D-set for the successively increasing values of k. Based on this method, we establish a relationship between γ(k−1)tg(G) and γktg(G), which, in turn, provides another upper bound on γktg(G).

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SHEIKHOLESLAMI, S. M., and L. VOLKMANN. "SIGNED TOTAL {K}-DOMINATION AND {K}-DOMATIC NUMBERS OF GRAPHS." Discrete Mathematics, Algorithms and Applications 04, no. 01 (March 2012): 1250006. http://dx.doi.org/10.1142/s1793830912500061.

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Let k be a positive integer, and let G be a simple graph with vertex set V(G). A function f : V(G) → {±1, ±2, …, ±k} is called a signed total {k}-dominating function if ∑u∈N(v) f(u) ≥ k for each vertex v ∈ V(G). A set {f1, f2, …, fd} of signed total {k}-dominating functions on G with the property that [Formula: see text] for each v∈V(G), is called a signed total {k}-dominating family (of functions) on G. The maximum number of functions in a signed total {k}-dominating family on G is the signed total {k}-domatic number of G, denoted by [Formula: see text]. Note that [Formula: see text] is the classical signed total domatic number dS(G). In this paper, we initiate the study of signed total k-domatic numbers in graphs, and we present some sharp upper bounds for [Formula: see text]. In addition, we determine [Formula: see text] for several classes of graphs. Some of our results are extensions of known properties of the signed total domatic number.

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Yang, Hong, Pu Wu, Sakineh Nazari-Moghaddam, Seyed Mahmoud Sheikholeslami, Xiaosong Zhang, Zehui Shao, and Yuan Yan Tang. "Bounds for signed double Roman k-domination in trees." RAIRO - Operations Research 53, no. 2 (April 2019): 627–43. http://dx.doi.org/10.1051/ro/2018043.

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Let k ≥ 1 be an integer and G be a simple and finite graph with vertex set V(G). A signed double Roman k-dominating function (SDRkDF) on a graph G is a function f:V(G) → {−1,1,2,3} such that (i) every vertex v with f(v) = −1 is adjacent to at least two vertices assigned a 2 or to at least one vertex w with f(w) = 3, (ii) every vertex v with f(v) = 1 is adjacent to at least one vertex w with f(w) ≥ 2 and (iii) ∑u∈N[v]f(u) ≥ k holds for any vertex v. The weight of a SDRkDF f is ∑u∈V(G) f(u), and the minimum weight of a SDRkDF is the signed double Roman k-domination number γksdR(G) of G. In this paper, we investigate the signed double Roman k-domination number of trees. In particular, we present lower and upper bounds on γksdR(T) for 2 ≤ k ≤ 6 and classify all extremal trees.

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6

Dehgardi, Nasrin, Maryam Atapour, and Abdollah Khodkar. "Twin signed k-domination numbers in directed graphs." Filomat 31, no. 20 (2017): 6367–78. http://dx.doi.org/10.2298/fil1720367d.

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Let D = (V;A) be a finite simple directed graph (digraph). A function f : V ? {-1,1} is called a twin signed k-dominating function (TSkDF) if f (N-[v]) ? k and f (N+[v]) ? k for each vertex v ? V. The twin signed k-domination number of D is ?* sk(D) = min{?(f)?f is a TSkDF of D}. In this paper, we initiate the study of twin signed k-domination in digraphs and present some bounds on ?* sk(D) in terms of the order, size and maximum and minimum indegrees and outdegrees, generalising some of the existing bounds for the twin signed domination numbers in digraphs and the signed k-domination numbers in graphs. In addition, we determine the twin signed k-domination numbers of some classes of digraphs.

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7

Kim, Kijung. "On k-rainbow domination in middle graphs." RAIRO - Operations Research 55, no. 6 (November 2021): 3447–58. http://dx.doi.org/10.1051/ro/2021163.

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Let G be a finite simple graph with vertex set V(G) and edge set E(G). A function f : V(G) → P({1,2,…,k}) is a k-rainbow dominating function on G if for each vertex v∈V(G) for which f(v) = ∅, it holds that ⋃u∈N(v) f(u) = {1,2,…,k}. The weight of a k-rainbow dominating function is the value ∑v∈V(G)|f(v)|. The k-rainbow domination number γrk (G) is the minimum weight of a k-rainbow dominating function on G. In this paper, we initiate the study of k-rainbow domination numbers in middle graphs. We define the concept of a middle k-rainbow dominating function, obtain some bounds related to it and determine the middle 3-rainbow domination number of some classes of graphs. We also provide upper and lower bounds for the middle 3-rainbow domination number of trees in terms of the matching number. In addition, we determine the 3-rainbow domatic number for the middle graph of paths and cycles.

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Shao, Zehui, Rija Erveš, Huiqin Jiang, Aljoša Peperko, Pu Wu, and Janez Žerovnik. "Double Roman Graphs in P(3k, k)." Mathematics 9, no. 4 (February 8, 2021): 336. http://dx.doi.org/10.3390/math9040336.

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A double Roman dominating function on a graph G=(V,E) is a function f:V→{0,1,2,3} with the properties that if f(u)=0, then vertex u is adjacent to at least one vertex assigned 3 or at least two vertices assigned 2, and if f(u)=1, then vertex u is adjacent to at least one vertex assigned 2 or 3. The weight of f equals w(f)=∑v∈Vf(v). The double Roman domination number γdR(G) of a graph G is the minimum weight of a double Roman dominating function of G. A graph is said to be double Roman if γdR(G)=3γ(G), where γ(G) is the domination number of G. We obtain the sharp lower bound of the double Roman domination number of generalized Petersen graphs P(3k,k), and we construct solutions providing the upper bounds, which gives exact values of the double Roman domination number for all generalized Petersen graphs P(3k,k). This implies that P(3k,k) is a double Roman graph if and only if either k≡0 (mod 3) or k∈{1,4}.

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9

Koam, Ali N. A., Ali Ahmad, Martin Bača, and Andrea Semaničová-Feňovčíková. "Modular edge irregularity strength of graphs." AIMS Mathematics 8, no. 1 (2022): 1475–87. http://dx.doi.org/10.3934/math.2023074.

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<abstract><p>For a simple graph $ G = (V, E) $ with the vertex set $ V(G) $ and the edge set $ E(G) $, a vertex labeling $ \varphi: V(G) \to \{1, 2, \dots, k\} $ is called a $ k $-labeling. The weight of an edge under the vertex labeling $ \varphi $ is the sum of the labels of its end vertices and the modular edge-weight is the remainder of the division of this sum by $ |E(G)| $. A vertex $ k $-labeling is called a modular edge irregular if for every two different edges their modular edge-weights are different. The maximal integer $ k $ minimized over all modular edge irregular $ k $-labelings is called the modular edge irregularity strength of $ G $. In the paper we estimate the bounds on the modular edge irregularity strength and for caterpillar, cycle, friendship graph and $ n $-sun we determine the precise values of this parameter that prove the sharpness of the lower bound.</p></abstract>

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10

Kuziak, Dorota, Iztok Peterin, and Ismael Yero. "Computing the (k-)monopoly number of direct product of graphs." Filomat 29, no. 5 (2015): 1163–71. http://dx.doi.org/10.2298/fil1505163k.

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Let G = (V,E) be a simple graph without isolated vertices and minimum degree ?(G), and let k ? {1-??(G)/2? ,..., ?(G)/2c?} be an integer. Given a set M ? V, a vertex v of G is said to be k-controlled by M if ?M(v)? ?(v)/2 + k where ?M(v) represents the quantity of neighbors v has in M and ?(v) the degree of v. The set M is called a k-monopoly if it k-controls every vertex v of G. The minimum cardinality of any k-monopoly is the k-monopoly number of G. In this article we study the k-monopoly number of direct product graphs. Specifically we obtain tight lower and upper bounds for the k-monopoly number of direct product graphs in terms of the k-monopoly numbers of its factors. Moreover, we compute the exact value for the k-monopoly number of several families of direct product graphs.

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11

Caadan, Julius Guhiting, Rolando N. Paluga, and Imelda S. Aniversario. "Upper Distance k-Cost Effective Number in the Join of Graphs." European Journal of Pure and Applied Mathematics 13, no. 3 (July 31, 2020): 701–9. http://dx.doi.org/10.29020/nybg.ejpam.v13i3.3657.

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Let k be a positive integer and G be a connected graph. The open k-neighborhoodset Nk G(v) of v ∈ V (G) is the set Nk G(v) = {u ∈ V (G) \ {v} : dG(u, v) ≤ k}. A set S of vertices of G is a distance k- cost effective if for every vertex u in S, |Nk G(u) ∩ Sc| − |NkG(u) ∩ S| ≥ 0. The maximum cardinality of a distance k- cost effective set of G is called the upper distance k- cost effective number of G. In this paper, we characterized a distance k- cost effective set in the join of two graphs. As direct consequences, the bounds or the exact values of the upper distance k- cost effective numbers are determined.

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12

Cheng, Rui, Gohar Ali, Gul Rahmat, Muhammad Yasin Khan, Andrea Semanicova-Fenovcikova, and Jia-Bao Liu. "Investigation of General Power Sum-Connectivity Index for Some Classes of Extremal Graphs." Complexity 2021 (August 20, 2021): 1–8. http://dx.doi.org/10.1155/2021/6623277.

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In this work, we introduce a new topological index called a general power sum-connectivity index and we discuss this graph invariant for some classes of extremal graphs. This index is defined by Y α G = ∑ u v ∈ E G d u d u + d v d v α , where d u and d v represent the degree of vertices u and v , respectively, and α ≥ 1 . A connected graph G is called a k -generalized quasi-tree if there exists a subset V k ⊂ V G of cardinality k such that the graph G − V k is a tree but for any subset V k − 1 ⊂ V G of cardinality k − 1 , the graph G − V k − 1 is not a tree. In this work, we find a sharp lower and some sharp upper bounds for this new sum-connectivity index.

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13

Harant, Jochen, and Stanislav Jendrol'. "Lightweight paths in graphs." Opuscula Mathematica 39, no. 6 (2019): 829–37. http://dx.doi.org/10.7494/opmath.2019.39.6.829.

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Let \(k\) be a positive integer, \(G\) be a graph on \(V(G)\) containing a path on \(k\) vertices, and \(w\) be a weight function assigning each vertex \(v\in V(G)\) a real weight \(w(v)\). Upper bounds on the weight \(w(P)=\sum_{v\in V(P)}w(v)\) of \(P\) are presented, where \(P\) is chosen among all paths of \(G\) on \(k\) vertices with smallest weight.

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14

Aghigh, Kamal, and Sudesh K. Khanduja. "ON THE MAIN INVARIANT OF ELEMENTS ALGEBRAIC OVER A HENSELIAN VALUED FIELD." Proceedings of the Edinburgh Mathematical Society 45, no. 1 (February 2002): 219–27. http://dx.doi.org/10.1017/s0013091500000936.

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AbstractLet $v$ be a henselian valuation of a field $K$ with value group $G$, let $\bar{v}$ be the (unique) extension of $v$ to a fixed algebraic closure $\bar{K}$ of $K$ and let $(\tilde{K},\tilde{v})$ be a completion of $(K,v)$. For $\alpha\in\bar{K}\setminus K$, let $M(\alpha,K)$ denote the set $\{\bar{v}(\alpha-\beta):\beta\in\bar{K},\ [K(\beta):K] \lt [K(\alpha):K]\}$. It is known that $M(\alpha,K)$ has an upper bound in $\bar{G}$ if and only if $[K(\alpha):K]=[\tilde{K}(\alpha):\tilde{K}]$, and that the supremum of $M(\alpha,K)$, which is denoted by $\delta_{K}(\alpha)$ (usually referred to as the main invariant of $\alpha$), satisfies a principle similar to the Krasner principle. Moreover, each complete discrete rank 1 valued field $(K,v)$ has the property that $\delta_{K}(\alpha)\in M(\alpha,K)$ for every $\alpha\in\bar{K}\setminus K$. In this paper the authors give a characterization of all those henselian valued fields $(K,v)$ which have the property mentioned above.AMS 2000 Mathematics subject classification: Primary 12J10; 12J25; 13A18

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Volkmann, Lutz. "Signed total k-independence in digraphs." Filomat 28, no. 10 (2014): 2121–30. http://dx.doi.org/10.2298/fil1410121v.

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Let k ? 2 be an integer. A function f:V(D) ? {-1,1} defined on the vertex set V(D) of a digraph D is a signed total k-independence function if ?x?N-(v)f(x) ? k - 1 for each v ? V(D), where N-(v) consists of all vertices of D from which arcs go into v. The weight of a signed total k-independence function f is defined by w(f)=?x?V(D)f(x). The maximum of weights w(f), taken over all signed total k-independence functions f on D, is the signed total k-independence number k?st(D) of D. In this work, we mainly present upper bounds on k?st(D), as for example k?st(D) ? n-2? ?- + 1-k)/2? and k?st(D)? ?+2k-?+-2/?+?+ ? n , where n is the order, ?- the maximum indegree and ?+ and ?+ are the maximum and minimum outdegree of the digraph D. Some of our results imply well-known properties on the signed total 2-independence number of graphs.

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Ghanem, Manal, Hasan Al-Ezeh, and Ala’a Dabbour. "Locating Chromatic Number of Powers of Paths and Cycles." Symmetry 11, no. 3 (March 18, 2019): 389. http://dx.doi.org/10.3390/sym11030389.

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Let c be a proper k-coloring of a graph G. Let π = { R 1 , R 2 , … , R k } be the partition of V ( G ) induced by c, where R i is the partition class receiving color i. The color code c π ( v ) of a vertex v of G is the ordered k-tuple ( d ( v , R 1 ) , d ( v , R 2 ) , … , d ( v , R k ) ) , where d ( v , R i ) is the minimum distance from v to each other vertex u ∈ R i for 1 ≤ i ≤ k . If all vertices of G have distinct color codes, then c is called a locating k-coloring of G. The locating-chromatic number of G, denoted by χ L ( G ) , is the smallest k such that G admits a locating coloring with k colors. In this paper, we give a characterization of the locating chromatic number of powers of paths. In addition, we find sharp upper and lower bounds for the locating chromatic number of powers of cycles.

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Beidar, Konstantin I., and Robert Wisbauer. "On uniform bounds of primeness in matrix rings." Journal of the Australian Mathematical Society 76, no. 2 (April 2004): 167–74. http://dx.doi.org/10.1017/s1446788700008879.

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AbstractA subset S of an associative ring R is a uniform insulator for R provided a S b ≠ 0 for any nonzero a, b ∈ R. The ring R is called uniformly strongly prime of bound m if R has uniform insulators and the smallest of those has cardinality m. Here we compute these bounds for matrix rings over fields and obtain refinements of some results of van den Berg in this context.More precisely, for a field F and a positive integer k, let m be the bound of the matrix ring Mk(F), and let n be dimF(V), where V is a subspace of Mk(F) of maximal dimension with respect to not containing rank one matrices. We show that m + n = k2. This implies, for example, that n = k2 − k if and only if there exists a (nonassociative) division algebra over F of dimension k.

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Amjadi, Jafar, Rana Khoeilar, N. Dehgardi, Lutz Volkmann, and S. M. Sheikholeslami. "The restrained rainbow bondage number of a graph." Tamkang Journal of Mathematics 49, no. 2 (June 30, 2018): 115–27. http://dx.doi.org/10.5556/j.tkjm.49.2018.2365.

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A restrained $k$-rainbow dominating function (R$k$RDF) of a graph $G$ is a function $f$ from the vertex set $V(G)$ to the set of all subsets of the set $\{1,2,\ldots,k\}$ such that for any vertex $v \in V (G)$ with $f(v) = \emptyset$ the conditions $\bigcup_{u \in N(v)} f(u)=\{1,2,\ldots,k\}$ and $|N(v)\cap \{u\in V\mid f(u)=\emptyset\}|\ge 1$ are fulfilled, where $N(v)$ is the open neighborhood of $v$. The weight of a restrained $k$-rainbow dominating function is the value $w(f)=\sum_{v\in V}|f (v)|$. The minimum weight of a restrained $k$-rainbow dominating function of $G$ is called the restrained $k$-rainbow domination number of $G$, denoted by $\gamma_{rrk}(G)$. The restrained $k$-rainbow bondage number $b_{rrk}(G)$ of a graph $G$ with maximum degree at least two is the minimum cardinality of all sets $F \subseteq E(G)$ for which $\gamma_{rrk}(G-F) > \gamma_{rrk}(G)$. In this paper, we initiate the study of the restrained $k$-rainbow bondage number in graphs and we present some sharp bounds for $b_{rr2}(G)$. In addition, we determine the restrained 2-rainbow bondage number of some classes of graphs.

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Cichacz, Sylwia, Agnieszka G�rlich, and Andrea Semani�ov�-Fe�ov��kov�. "Upper bounds on distance vertex irregularity strength of some families of graphs." Opuscula Mathematica 42, no. 4 (2022): 561–71. http://dx.doi.org/10.7494/opmath.2022.42.4.561.

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For a graph \(G\) its distance vertex irregularity strength is the smallest integer \(k\) for which one can find a labeling \(f: V(G)\to \{1, 2, \dots, k\}\) such that \[ \sum_{x\in N(v)}f(x)\neq \sum_{x\in N(u)}f(x)\] for all vertices \(u,v\) of \(G\), where \(N(v)\) is the open neighborhood of \(v\). In this paper we present some upper bounds on distance vertex irregularity strength of general graphs. Moreover, we give upper bounds on distance vertex irregularity strength of hypercubes and trees.

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Sun, Yuefang, and Gregory Gutin. "Strong Subgraph Connectivity of Digraphs." Graphs and Combinatorics 37, no. 3 (March 18, 2021): 951–70. http://dx.doi.org/10.1007/s00373-021-02294-w.

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AbstractLet $$D=(V,A)$$ D = ( V , A ) be a digraph of order n, S a subset of V of size k and $$2\le k\le n$$ 2 ≤ k ≤ n . A strong subgraph H of D is called an S-strong subgraph if $$S\subseteq V(H)$$ S ⊆ V ( H ) . A pair of S-strong subgraphs $$D_1$$ D 1 and $$D_2$$ D 2 are said to be arc-disjoint if $$A(D_1)\cap A(D_2)=\emptyset$$ A ( D 1 ) ∩ A ( D 2 ) = ∅ . A pair of arc-disjoint S-strong subgraphs $$D_1$$ D 1 and $$D_2$$ D 2 are said to be internally disjoint if $$V(D_1)\cap V(D_2)=S$$ V ( D 1 ) ∩ V ( D 2 ) = S . Let $$\kappa _S(D)$$ κ S ( D ) (resp. $$\lambda _S(D)$$ λ S ( D ) ) be the maximum number of internally disjoint (resp. arc-disjoint) S-strong subgraphs in D. The strong subgraphk-connectivity is defined as $$\begin{aligned} \kappa _k(D)=\min \{\kappa _S(D)\mid S\subseteq V, |S|=k\}. \end{aligned}$$ κ k ( D ) = min { κ S ( D ) ∣ S ⊆ V , | S | = k } . As a natural counterpart of the strong subgraph k-connectivity, we introduce the concept of strong subgraphk-arc-connectivity which is defined as $$\begin{aligned} \lambda _k(D)=\min \{\lambda _S(D)\mid S\subseteq V(D), |S|=k\}. \end{aligned}$$ λ k ( D ) = min { λ S ( D ) ∣ S ⊆ V ( D ) , | S | = k } . A digraph $$D=(V, A)$$ D = ( V , A ) is called minimally strong subgraph$$(k,\ell )$$ ( k , ℓ ) -(arc-)connected if $$\kappa _k(D)\ge \ell$$ κ k ( D ) ≥ ℓ (resp. $$\lambda _k(D)\ge \ell$$ λ k ( D ) ≥ ℓ ) but for any arc $$e\in A$$ e ∈ A , $$\kappa _k(D-e)\le \ell -1$$ κ k ( D - e ) ≤ ℓ - 1 (resp. $$\lambda _k(D-e)\le \ell -1$$ λ k ( D - e ) ≤ ℓ - 1 ). In this paper, we first give complexity results for $$\lambda _k(D)$$ λ k ( D ) , then obtain some sharp bounds for the parameters $$\kappa _k(D)$$ κ k ( D ) and $$\lambda _k(D)$$ λ k ( D ) . Finally, minimally strong subgraph $$(k,\ell )$$ ( k , ℓ ) -connected digraphs and minimally strong subgraph $$(k,\ell )$$ ( k , ℓ ) -arc-connected digraphs are studied.

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Gharibyan, Aram H., and Petros A. Petrosyan. "LOCALLY-BALANCED $k$-PARTITIONS OF GRAPHS." Proceedings of the YSU A: Physical and Mathematical Sciences 55, no. 2 (255) (June 18, 2021): 96–112. http://dx.doi.org/10.46991/pysu:a/2021.55.2.096.

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In this paper we generalize locally-balanced $2$-partitions of graphs and introduce a new notion, the locally-balanced $k$-partitions of graphs, defined as follows: a $k$-partition of a graph $G$ is a surjection $f:V(G)\rightarrow \{0,1,\ldots,k-1\}$. A $k$-partition ($k\geq 2$) $f$ of a graph $G$ is a locally-balanced with an open neighborhood, if for every $v\in V(G)$ and any $0\leq i<j\leq k-1$ $$\left\vert \vert \{u\in N_{G}(v)\colon\,f(u)=i\}\vert - \vert \{u\in N_{G}(v)\colon\,f(u)=j\}\vert \right\vert\leq 1.$$ A $k$-partition ($k\geq 2$) $f^{\prime}$ of a graph $G$ is a locally-balanced with a closed neighborhood, if for every $v\in V(G)$ and any $0\leq i<j\leq k-1$ $$\left\vert \vert \{u\in N_{G}[v]\colon\,f^{\prime}(u)=i\}\vert - \vert \{u\in N_{G}[v]\colon\,f^{\prime}(u)=j\}\vert \right\vert\leq 1.$$ The minimum number $k$ ($k\geq 2$), for which a graph $G$ has a locally-balanced $k$-partition with an open (a closed) neighborhood, is called an $lb$-open ($lb$-closed) chromatic number of $G$ and denoted by $\chi_{(lb)}(G)$ ($\chi_{[lb]}(G)$). In this paper we determine or bound the $lb$-open and $lb$-closed chromatic numbers of several families of graphs. We also consider the connections of $lb$-open and $lb$-closed chromatic numbers of graphs with other chromatic numbers such as injective and $2$-distance chromatic numbers.

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Buyantogtokh, Lkhagva, Batmend Horoldagva, and Kinkar Chandra Das. "On General Reduced Second Zagreb Index of Graphs." Mathematics 10, no. 19 (September 29, 2022): 3553. http://dx.doi.org/10.3390/math10193553.

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Graph-based molecular structure descriptors (often called “topological indices”) are useful for modeling the physical and chemical properties of molecules, designing pharmacologically active compounds, detecting environmentally hazardous substances, etc. The graph invariant GRMα, known under the name general reduced second Zagreb index, is defined as GRMα(Γ)=∑uv∈E(Γ)(dΓ(u)+α)(dΓ(v)+α), where dΓ(v) is the degree of the vertex v of the graph Γ and α is any real number. In this paper, among all trees of order n, and all unicyclic graphs of order n with girth g, we characterize the extremal graphs with respect to GRMα(α≥−12). Using the extremal unicyclic graphs, we obtain a lower bound on GRMα(Γ) of graphs in terms of order n with k cut edges, and completely determine the corresponding extremal graphs. Moreover, we obtain several upper bounds on GRMα of different classes of graphs in terms of order n, size m, independence number γ, chromatic number k, etc. In particular, we present an upper bound on GRMα of connected triangle-free graph of order n>2, m>0 edges with α>−1.5, and characterize the extremal graphs. Finally, we prove that the Turán graph Tn(k) gives the maximum GRMα(α≥−1) among all graphs of order n with chromatic number k.

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23

Wilson, George V. "Mixed Groups of Finite Nilstufe." Canadian Mathematical Bulletin 30, no. 2 (June 1, 1987): 255–56. http://dx.doi.org/10.4153/cmb-1987-036-7.

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AbstractThis paper constructs a class of examples to show that for torsion-free groups H with finite nilstufe v(H) = n < ∞ there can be divisible torsion groups D with v(H ⊕ D) - n + k for all k ≤ n + 1. This answers a question of Feigelstock. The construction is based on a proposition which bounds v(H ⊕ D) in terms of v(H) and rank (D).

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Salindeho, Brilly Maxel, Hilda Assiyatun, and Edy Tri Baskoro. "On The Locating-Chromatic Numbers of Subdivisions of Friendship Graph." Journal of the Indonesian Mathematical Society 26, no. 2 (July 10, 2020): 175–84. http://dx.doi.org/10.22342/jims.26.2.822.175-184.

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Let c be a k-coloring of a connected graph G and let pi={C1,C2,...,Ck} be the partition of V(G) induced by c. For every vertex v of G, let c_pi(v) be the coordinate of v relative to pi, that is c_pi(v)=(d(v,C1 ),d(v,C2 ),...,d(v,Ck )), where d(v,Ci )=min{d(v,x)|x in Ci }. If every two vertices of G have different coordinates relative to pi, then c is said to be a locating k-coloring of G. The locating-chromatic number of G, denoted by chi_L (G), is the least k such that there exists a locating k-coloring of G. In this paper, we determine the locating-chromatic numbers of some subdivisions of the friendship graph Fr_t, that is the graph obtained by joining t copies of 3-cycle with a common vertex, and we give lower bounds to the locating-chromatic numbers of few other subdivisions of Fr_t.

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Lim, Seonhee, Nicolas de Saxcé, and Uri Shapira. "Dimension Bound for Badly Approximable Grids." International Mathematics Research Notices 2019, no. 20 (January 26, 2018): 6317–46. http://dx.doi.org/10.1093/imrn/rnx330.

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Abstract We show that there exists a subset of full Lebesgue measure $V\subset \mathbb{R}^{n}$ such that for every ϵ > 0 there exists δ > 0 such that for any v ∈ V the dimension of the set of vectors w satisfying $$ \liminf_{k\to\infty} k^{1/n}\langle kv-w\rangle\geqslant \epsilon$$ (where 〈⋅〉 denotes the distance from the nearest integer) is bounded above by n − δ. This result is obtained as a corollary of a discussion in homogeneous dynamics and the main tool in the proof is a relative version of the principle of uniqueness of measures with maximal entropy.

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26

Kohlenbach, Ulrich. "Relative constructivity." Journal of Symbolic Logic 63, no. 4 (December 1998): 1218–38. http://dx.doi.org/10.2307/2586648.

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In a previous paper [13] we introduced a hierarchy (GnAω)n∈ℕ of subsystems of classical arithmetic in all finite types where the growth of definable functions of GnAω corresponds to the well-known Grzegorczyk hierarchy. Let AC-qf denote the schema of quantifier-free choice.[11], [13], [8] and [7] study various analytical principles Γ in the context of the theories GnAω + AC-qf (mainly for n = 2) and use proof-theoretic tools like, e.g., monotone functional interpretation (which was introduced in [12]) to determine their impact on the growth of uniform bounds Φ such thatwhich are extractable from given proofs (based on these principles Γ) of sentencesHere A0(u, k, v, w) is quantifier-free and contains only u, k, v, w as free variables; t is a closed term and ≤p is defined pointwise. The term ‘uniform bound’ refers to the fact that Φ does not depend on v ≤ptuk (see [12] for the relevance of such uniform bounds in numerical analysis and for concrete applications to approximation theory).

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Barišić, Ana Klobučar, and Antoaneta Klobučar. "Double total domination number in certain chemical graphs." AIMS Mathematics 7, no. 11 (2022): 19629–40. http://dx.doi.org/10.3934/math.20221076.

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<abstract><p>Let $ G $ be a graph with the vertex set $ V(G) $. A set $ D\subseteq V(G) $ is a total k-dominating set if every vertex $ v\in V(G) $ has at least $ k $ neighbours in $ D $. The total k-domination number $ \gamma_{kt}(G) $ is the cardinality of the smallest total k-dominating set. For $ k = 2 $ the total 2-dominating set is called double total dominating set. In this paper we determine the upper and lower bounds and some exact values for double total domination number on pyrene network $ PY(n) $, $ n\geq 1 $ and hexabenzocoronene $ XC(n) $ $ n\geq 2 $, where pyrene network and hexabenzocoronene are composed of congruent hexagons.</p></abstract>

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28

Galatenko, A. V., and V. A. Kuzovikhina. "A Model of Secure Functioning of Computer Systems." Programmnaya Ingeneria 12, no. 3 (May 19, 2021): 150–56. http://dx.doi.org/10.17587/prin.12.150-156.

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We propose an automata model of computer system security. A system is represented by a finite automaton with states partitioned into two subsets: "secure" and "insecure". System functioning is secure if the number of consecutive insecure states is not greater than some nonnegative integer k. This definition allows one to formally reflect responsiveness to security breaches. The number of all input sequences that preserve security for the given value of k is referred to as a k-secure language. We prove that if a language is k-secure for some natural and automaton V, then it is also k-secure for any 0 < k < k and some automaton V = V (k). Reduction of the value of k is performed at the cost of amplification of the number of states. On the other hand, for any non-negative integer k there exists a k-secure language that is not k"-secure for any natural k" > k. The problem of reconstruction of a k-secure language using a conditional experiment is split into two subcases. If the cardinality of an input alphabet is bound by some constant, then the order of Shannon function of experiment complexity is the same for al k; otherwise there emerges a lower bound of the order nk.

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Bent-Usman, Wardah Masanggila, Rowena Isla, and Sergio Canoy. "Neighborhood Connected k-Fair Domination Under Some Binary Operations." European Journal of Pure and Applied Mathematics 12, no. 3 (August 2, 2019): 1337–49. http://dx.doi.org/10.29020/nybg.ejpam.v12i3.3506.

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Let G=(V(G),E(G)) be a simple graph. A neighborhood connected k-fair dominating set (nckfd-set) is a dominating set S subset V(G) such that |N(u) intersection S|=k for every u is an element of V(G)\S and the induced subgraph of S is connected. In this paper, we introduce and invistigate the notion of neighborhood connected k-fair domination in graphs. We also characterize such dominating sets in the join, corona, lexicographic and cartesians products of graphs and determine the exact value or sharp bounds of their corresponding neighborhood connected k-fair domination number.

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30

SUSANTO, FAISAL, KRISTIANA WIJAYA, PRASANTI MIA PURNAMA, and SLAMIN S. "On Distance Irregular Labeling of Disconnected Graphs." Kragujevac Journal of Mathematics 46, no. 4 (August 2022): 507–23. http://dx.doi.org/10.46793/kgjmat2204.507s.

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A distance irregular k-labeling of a graph G is a function f : V (G) → {1, 2, . . . , k} such that the weights of all vertices are distinct. The weight of a vertex v, denoted by wt(v), is the sum of labels of all vertices adjacent to v (distance 1 from v), that is, wt(v) = P u∈N(v) f(u). If the graph G admits a distance irregular labeling then G is called a distance irregular graph. The distance irregularity strength of G is the minimum k for which G has a distance irregular k-labeling and is denoted by dis(G). In this paper, we derive a new lower bound of distance irregularity strength for graphs with t pendant vertices. We also determine the distance irregularity strength of some families of disconnected graphs namely disjoint union of paths, suns, helms and friendships.

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31

Khemmani, Varanoot, and Supachoke Isariyapalakul. "The Multiresolving Sets of Graphs with Prescribed Multisimilar Equivalence Classes." International Journal of Mathematics and Mathematical Sciences 2018 (August 1, 2018): 1–6. http://dx.doi.org/10.1155/2018/8978193.

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For a set W=w1,w2,…,wk of vertices and a vertex v of a connected graph G, the multirepresentation of v with respect to W is the k-multiset mr(v∣W)=dv,w1,dv,w2,…,dv,wk, where d(v,wi) is the distance between the vertices v and wi for i=1,2,…,k. The set W is a multiresolving set of G if every two distinct vertices of G have distinct multirepresentations with respect to W. The minimum cardinality of a multiresolving set of G is the multidimension dimM(G) of G. It is shown that, for every pair k,n of integers with k≥3 and n≥3(k-1), there is a connected graph G of order n with dimM(G)=k. For a multiset {a1,a2,…,ak} and an integer c, we define {a1,a2,…,ak}+c,c,…,c=a1+c,a2+c,…,ak+c. A multisimilar equivalence relation RW on V(G) with respect to W is defined by u RW v if mr(u∣W)=mrv∣W+cWu,v,cWu,v,…,cWu,v for some integer cW(u,v). We study the relationship between the elements in multirepresentations of vertices that belong to the same multisimilar equivalence class and also establish the upper bound for the cardinality of a multisimilar equivalence class. Moreover, a multiresolving set with prescribed multisimilar equivalence classes is presented.

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32

Bertolo, Riccardo, Iliya Bluskov, and Heikki Hämäläinen. "Upper bounds on the general covering numberCλ(v,k,t,m)." Journal of Combinatorial Designs 12, no. 5 (2004): 362–80. http://dx.doi.org/10.1002/jcd.20019.

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33

CZYGRINOW, ANDRZEJ, LOUIS DeBIASIO, H. A. KIERSTEAD, and THEODORE MOLLA. "An Extension of the Hajnal–Szemerédi Theorem to Directed Graphs." Combinatorics, Probability and Computing 24, no. 5 (October 28, 2014): 754–73. http://dx.doi.org/10.1017/s0963548314000716.

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Hajnal and Szemerédi proved that every graph G with |G| = ks and δ(G)⩾ k(s − 1) contains k disjoint s-cliques; moreover this degree bound is optimal. We extend their theorem to directed graphs by showing that every directed graph $\vv G$ with |$\vv G$| = ks and δ($\vv G$) ⩾ 2k(s − 1) − 1 contains k disjoint transitive tournaments on s vertices, where δ($\vv G$)= minv∈V($\vv G$)d−(v)+d+(v). Our result implies the Hajnal–Szemerédi theorem, and its degree bound is optimal. We also make some conjectures regarding even more general results for multigraphs and partitioning into other tournaments. One of these conjectures is supported by an asymptotic result.

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34

Kou, Zheng, Saeed Kosari, Guoliang Hao, Jafar Amjadi, and Nesa Khalili. "Quadruple Roman Domination in Trees." Symmetry 13, no. 8 (July 22, 2021): 1318. http://dx.doi.org/10.3390/sym13081318.

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This paper is devoted to the study of the quadruple Roman domination in trees, and it is a contribution to the Special Issue “Theoretical computer science and discrete mathematics” of Symmetry. For any positive integer k, a [k]-Roman dominating function ([k]-RDF) of a simple graph G is a function from the vertex set V of G to the set {0,1,2,…,k+1} if for any vertex u∈V with f(u)<k, ∑x∈N(u)∪{u}f(x)≥|{x∈N(u):f(x)≥1}|+k, where N(u) is the open neighborhood of u. The weight of a [k]-RDF is the value Σv∈Vf(v). The minimum weight of a [k]-RDF is called the [k]-Roman domination number γ[kR](G) of G. In this paper, we establish sharp upper and lower bounds on γ[4R](T) for nontrivial trees T and characterize extremal trees.

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35

Wang, Shaohui, Zehui Shao, Jia-Bao Liu, and Bing Wei. "The Bounds of Vertex Padmakar–Ivan Index on k-Trees." Mathematics 7, no. 4 (April 1, 2019): 324. http://dx.doi.org/10.3390/math7040324.

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The Padmakar–Ivan ( P I ) index is a distance-based topological index and a molecular structure descriptor, which is the sum of the number of vertices over all edges u v of a graph such that these vertices are not equidistant from u and v. In this paper, we explore the results of P I -indices from trees to recursively clustered trees, the k-trees. Exact sharp upper bounds of PI indices on k-trees are obtained by the recursive relationships, and the corresponding extremal graphs are given. In addition, we determine the P I -values on some classes of k-trees and compare them, and our results extend and enrich some known conclusions.

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36

Chen, Danmei. "The kth Local Exponent of Doubly Symmetric Primitive Digraphs with d Loops." Symmetry 14, no. 8 (August 7, 2022): 1623. http://dx.doi.org/10.3390/sym14081623.

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Let D be a primitive digraph of order n. The exponent of a vertex x in V(D) is denoted γD(x), which is the smallest integer q such that for any vertex y, there is a walk of length q from x to y. Let V(D)={v1,v2,⋯,vn}. We order the vertices of V(D) so that γD(v1)≤γD(v2)≤⋯≤γD(vn) is satisfied. Then, for any integer k satisfying 1≤k≤n,γD(vk) is called the kth local exponent of D and is denoted by expD(k). Let DSn(d) represent the set of all doubly symmetric primitive digraphs with n vertices and d loops, where d is an integer such that 1≤d≤n. In this paper, we determine the upper bound for the kth local exponent of DSn(d), where 1≤k≤n. In addition, we find that the upper bound for the kth local exponent of DSn(d) can be reached, where 1≤k≤n.

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37

Mohanapriya, N., K. Kalaiselvi, V. Aparna, Dafik, and I. H. Agustin. "On r-dynamic coloring of central vertex join of path, cycle with certain graphs." Journal of Physics: Conference Series 2157, no. 1 (January 1, 2022): 012007. http://dx.doi.org/10.1088/1742-6596/2157/1/012007.

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Abstract Let G = (V, E) be a simple finite connected and undirected graph with n vertices and m edges. The n vertices are assigned the colors through mapping c : V [G] → I +. An r-dynamic coloring is a proper k-coloring of a graph G such that each vertex of G receive colors in at least min{deg(υ),r} different color classes. The minimum k such that the graph G has r-dynamic k coloring is called the r-dynamic chromatic number of graph G denoted as χ r (G). Let G 1 and G 2 be a graphs with n 1 and n 2 vertices and m 1 and m 2 edges. The central vertex join of G1 and G 2 is the graph G 1 V ˙ G 2 is obtained from C(G 1) and G 2 joining each vertex of G 1 with every vertex of G 2. The aim of this paper is to obtain the lower bound for r-dynamic chromatic number of central vertex join of path with a graph G, central vertex join of cycle with a graph G and r-dynamic chromatic number of P m V ˙ P n , P m V ˙ K n , P m V ˙ K n , P m V ˙ C n , C m V ˙ K n and C m V ˙ C n respectively.

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38

Shao, Zehui, S. M. Sheikholeslami, Pu Wu, and Jia-Biao Liu. "The Metric Dimension of Some Generalized Petersen Graphs." Discrete Dynamics in Nature and Society 2018 (August 1, 2018): 1–10. http://dx.doi.org/10.1155/2018/4531958.

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The distance d(u,v) between two distinct vertices u and v in a graph G is the length of a shortest (u,v)-path in G. For an ordered subset W={w1,w2,…,wk} of vertices and a vertex v in G, the code of v with respect to W is the ordered k-tuple cW(v)=(d(v,w1),d(v,w2),…,d(v,wk)). The set W is a resolving set for G if every two vertices of G have distinct codes. The metric dimension of G is the minimum cardinality of a resolving set of G. In this paper, we first extend the results of the metric dimension of P(n,3) and P(n,4) and study bounds on the metric dimension of the families of the generalized Petersen graphs P(2k,k) and P(3k,k). The obtained results mean that these families of graphs have constant metric dimension.

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39

Wang, Zhao, Yaping Mao, Kinkar Chandra Das, and Yilun Shang. "Nordhaus–Gaddum-Type Results for the Steiner Gutman Index of Graphs." Symmetry 12, no. 10 (October 16, 2020): 1711. http://dx.doi.org/10.3390/sym12101711.

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Building upon the notion of the Gutman index SGut(G), Mao and Das recently introduced the Steiner Gutman index by incorporating Steiner distance for a connected graph G. The Steiner Gutman k-index SGutk(G) of G is defined by SGutk(G)=∑S⊆V(G),|S|=k∏v∈SdegG(v)dG(S), in which dG(S) is the Steiner distance of S and degG(v) is the degree of v in G. In this paper, we derive new sharp upper and lower bounds on SGutk, and then investigate the Nordhaus-Gaddum-type results for the parameter SGutk. We obtain sharp upper and lower bounds of SGutk(G)+SGutk(G¯) and SGutk(G)·SGutk(G¯) for a connected graph G of order n, m edges, maximum degree Δ and minimum degree δ.

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40

Raza, Hassan, Jia-Bao Liu, Muhammad Azeem, and Muhammad Faisal Nadeem. "Partition Dimension of Generalized Petersen Graph." Complexity 2021 (October 29, 2021): 1–14. http://dx.doi.org/10.1155/2021/5592476.

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Let G = V G , E G be the connected graph. For any vertex i ∈ V G and a subset B ⊆ V G , the distance between i and B is d i ; B = min d i , j | j ∈ B . The ordered k -partition of V G is Π = B 1 , B 2 , … , B k . The representation of vertex i with respect to Π is the k -vector, that is, r i | Π = d i , B 1 , d i , B 2 , … , d i , B k . The partition Π is called the resolving (distinguishing) partition if r i | Π ≠ r j | Π , for all distinct i , j ∈ V G . The minimum cardinality of the resolving partition is called the partition dimension, denoted as pd G . In this paper, we consider the upper bound for the partition dimension of the generalized Petersen graph in terms of the cardinalities of its partite sets.

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41

Bluskov, I., M. Greig, and K. Heinrich. "Infinite Classes of Covering Numbers." Canadian Mathematical Bulletin 43, no. 4 (December 1, 2000): 385–96. http://dx.doi.org/10.4153/cmb-2000-046-9.

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AbstractLet D be a family of k-subsets (called blocks) of a v-set X(v). Then D is a (v, k, t) covering design or covering if every t-subset of X(v) is contained in at least one block of D. The number of blocks is the size of the covering, and the minimum size of the covering is called the covering number. In this paper we consider the case t = 2, and find several infinite classes of covering numbers. We also give upper bounds on other classes of covering numbers.

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42

Sheikholeslami, S. M., and L. Volkmann. "Signed star (j, k)-domatic numbers of digraphs." Discrete Mathematics, Algorithms and Applications 07, no. 02 (May 25, 2015): 1550006. http://dx.doi.org/10.1142/s1793830915500068.

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Let D be a simple digraph with arc set A(D), and let j and k be two positive integers. A function f : A(D) → {-1, 1} is said to be a signed star j-dominating function (SSjDF) on D if ∑a ∈ A(v) f(a) ≥ j for every vertex v of D, where A(v) is the set of arcs with head v. A set {f1, f2, …, fd} of distinct SSjDFs on D with the property that [Formula: see text] for each a ∈ A(D), is called a signed star (j, k)-dominating (SS(j, k)D) family (of functions) on D. The maximum number of functions in a SS(j, k)D family on D is the signed star (j, k)-domatic number of D, denoted by [Formula: see text]. In this paper, we study properties of the signed star (j, k)-domatic number of a digraph D. In particular, we determine bounds on [Formula: see text]. Some of our results extend these ones given by Sheikholeslami and Volkmann [Signed star k-domination and k-domatic number of digraphs, submitted] for the signed (j, 1)-domatic number.

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Zhang, Ke, Haixing Zhao, Zhonglin Ye, Yu Zhu, and Liang Wei. "The Bounds of the Edge Number in Generalized Hypertrees." Mathematics 7, no. 1 (December 20, 2018): 2. http://dx.doi.org/10.3390/math7010002.

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A hypergraph H = ( V , ε ) is a pair consisting of a vertex set V , and a set ε of subsets (the hyperedges of H ) of V . A hypergraph H is r -uniform if all the hyperedges of H have the same cardinality r . Let H be an r -uniform hypergraph, we generalize the concept of trees for r -uniform hypergraphs. We say that an r -uniform hypergraph H is a generalized hypertree ( G H T ) if H is disconnected after removing any hyperedge E , and the number of components of G H T − E is a fixed value k ( 2 ≤ k ≤ r ) . We focus on the case that G H T − E has exactly two components. An edge-minimal G H T is a G H T whose edge set is minimal with respect to inclusion. After considering these definitions, we show that an r -uniform G H T on n vertices has at least 2 n / ( r + 1 ) edges and it has at most n − r + 1 edges if r ≥ 3 and n ≥ 3 , and the lower and upper bounds on the edge number are sharp. We then discuss the case that G H T − E has exactly k ( 2 ≤ k ≤ r − 1 ) components.

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44

Mageshwaran, K., G. Kalaimurugan, Bussakorn Hammachukiattikul, Vediyappan Govindan, and Ismail Naci Cangul. "On L h , k -Labeling Index of Inverse Graphs Associated with Finite Cyclic Groups." Journal of Mathematics 2021 (March 25, 2021): 1–7. http://dx.doi.org/10.1155/2021/5583433.

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An L h , k -labeling of a graph G = V , E is a function f : V ⟶ 0 , ∞ such that the positive difference between labels of the neighbouring vertices is at least h and the positive difference between the vertices separated by a distance 2 is at least k . The difference between the highest and lowest assigned values is the index of an L h , k -labeling. The minimum number for which the graph admits an L h , k -labeling is called the required possible index of L h , k -labeling of G , and it is denoted by λ k h G . In this paper, we obtain an upper bound for the index of the L h , k -labeling for an inverse graph associated with a finite cyclic group, and we also establish the fact that the upper bound is sharp. Finally, we investigate a relation between L h , k -labeling with radio labeling of an inverse graph associated with a finite cyclic group.

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45

XU, BAOGANG, and XINGXING YU. "Better Bounds for k-Partitions of Graphs." Combinatorics, Probability and Computing 20, no. 4 (May 31, 2011): 631–40. http://dx.doi.org/10.1017/s0963548311000204.

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Let G be a graph with m edges, and let k be a positive integer. We show that V(G) admits a k-partition V1, . . . Vk such that $e(V_i)\leq \frac 1{k^2}m+\frac {k-1}{2k^2}(\sqrt{2m+1/4}-1/2)$ for i ∈ {1, 2, . . . k}, and $e(V_1, \ldots, V_k)\geq \frac{k-1}{ k} m +\frac{k-1}{ 2k}\sqrt{2m+1/4} +O(k)$, where e(Vi) denotes the number of edges with both ends in Vi and $e(V_1,\ldots, V_k)=m-\sum_{i=1}^ke(V_i)$. This answers a problem of Bollobás and Scott [2] in the affirmative. Moreover, $\binom{k+1}{ 2}e(V_i)+\frac k2\sum_{j\ne i}e(V_j)\le m + O(k^2)$ for i ∈ {1, 2, . . ., k}, which is close to being best possible and settles another problem of Bollobás and Scott [2].

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46

Rana, A. "On the Total Vertex Irregular Labeling of Proper Interval Graphs." Journal of Scientific Research 12, no. 4 (September 1, 2020): 537–43. http://dx.doi.org/10.3329/jsr.v12i4.45923.

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A labeling of a graph is a mapping that maps some set of graph elements to a set of numbers (usually positive integers). For a simple graph G = (V, E) with vertex set V and edge set E, a labeling Φ: V ∪ E → {1, 2, ..., k} is called total k-labeling. The associated vertex weight of a vertex x∈ V under a total k-labeling Φ is defined as wt(x) = Φ(x) + ∑y∈N(x) Φ(xy) where N(x) is the set of neighbors of the vertex x. A total k-labeling is deﬁned to be a vertex irregular total labeling of a graph, if for every two different vertices x and y of G, wt(x)≠wt(y). The minimum k for which a graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G, tvs(G). In this paper, total vertex irregularity strength of interval graphs is studied. In particular, an efficient algorithm is designed to compute tvs of proper interval graphs and bounds of tvs is presented for interval graphs.

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Rupnik Poklukar, Darja, and Janez Žerovnik. "On the Double Roman Domination in Generalized Petersen Graphs P(5k,k)." Mathematics 10, no. 1 (January 1, 2022): 119. http://dx.doi.org/10.3390/math10010119.

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A double Roman dominating function on a graph G=(V,E) is a function f:V→{0,1,2,3} satisfying the condition that every vertex u for which f(u)=0 is adjacent to at least one vertex assigned 3 or at least two vertices assigned 2, and every vertex u with f(u)=1 is adjacent to at least one vertex assigned 2 or 3. The weight of f equals w(f)=∑v∈Vf(v). The double Roman domination number γdR(G) of a graph G equals the minimum weight of a double Roman dominating function of G. We obtain closed expressions for the double Roman domination number of generalized Petersen graphs P(5k,k). It is proven that γdR(P(5k,k))=8k for k≡2,3mod5 and 8k≤γdR(P(5k,k))≤8k+2 for k≡0,1,4mod5. We also improve the upper bounds for generalized Petersen graphs P(20k,k).

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48

Shaheen, Ramy, Suhail Mahfud, and Ali Kassem. "Irreversible k -Threshold Conversion Number of Circulant Graphs." Journal of Applied Mathematics 2022 (August 11, 2022): 1–14. http://dx.doi.org/10.1155/2022/1250951.

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An irreversible conversion process is a dynamic process on a graph where a one-way change of state (from state 0 to state 1) is applied on the vertices if they satisfy a conversion rule that is determined at the beginning of the study. The irreversible k -threshold conversion process on a graph G = V , E is an iterative process which begins by choosing a set S 0 ⊆ V , and for each step t t = 1 , 2 , ⋯ , , S t is obtained from S t − 1 by adjoining all vertices that have at least k neighbors in S t − 1 . S 0 is called the seed set of the k -threshold conversion process, and if S t = V G for some t ≥ 0 , then S 0 is an irreversible k -threshold conversion set (IkCS) of G . The k -threshold conversion number of G (denoted by ( C k G ) is the minimum cardinality of all the IkCSs of G . In this paper, we determine C 2 G for the circulant graph C n 1 , r when r is arbitrary; we also find C 3 C n 1 , r when r = 2 , 3 . We also introduce an upper bound for C 3 C n 1 , 4 . Finally, we suggest an upper bound for C 3 C n 1 , r if n ≥ 2 r + 1 and n ≡ 0 mod 2 r + 1 .

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49

Matala-aho, Tapani. "On Diophantine approximations of the solutions of q-functional equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 132, no. 3 (June 2002): 639–59. http://dx.doi.org/10.1017/s0308210500001827.

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Given a sequence of linear forms in m ≥ 2 complex or p-adic numbers α1, …,αm ∈ Kv with appropriate growth conditions, Nesterenko proved a lower bound for the dimension d of the vector space Kα1 + ··· + Kαm over K, when K = Q and v is the infinite place. We shall generalize Nesterenko's dimension estimate over number fields K with appropriate places v, if the lower bound condition for |Rn| is replaced by the determinant condition. For the q-series approximations also a linear independence measure is given for the d linearly independent numbers. As an application we prove that the initial values F(t), F(qt), …, F(qm−1t) of the linear homogeneous q-functional equation where N = N(q, t), Pi = Pi(q, t) ∈ K[q, t] (i = 1, …, m), generate a vector space of dimension d ≥ 2 over K under some conditions for the coefficient polynomials, the solution F(t) and t, q ∈ K*.

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50

Venkateswarlu, Bolineni, Pinninti Thirupathi Reddy, Settipalli Sridevi, and Vaishnavy Sujatha. "A Certain Subclass of Analytic Functions with Negative Coefficients Defined by Gegenbauer Polynomials." Tatra Mountains Mathematical Publications 78, no. 1 (October 1, 2021): 73–84. http://dx.doi.org/10.2478/tmmp-2021-0006.

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Abstract In this paper, we introduce a new subclass of analytic functions with negative coefficients defined by Gegenbauer polynomials. We obtain coefficient bounds, growth and distortion properties, extreme points and radii of starlikeness, convexity and close-to-convexity for functions belonging to the class T S λ m ( γ , e , k , v ) TS_\lambda ^m(\gamma ,e,k,v) . Furthermore, we obtained the Fekete-Szego problem for this class.

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