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Zeitschriftenartikel zum Thema „V.R.G. Papier“

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Veröffentlicht am 25. Juli 2024

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1

CHEN, LI, und TONGSUO WU. „ON RINGS R WHOSE GRAPHS Γ(R) SATISFY CONDITION (Kp)“. Journal of Algebra and Its Applications 10, Nr. 04 (August 2011): 665–74. http://dx.doi.org/10.1142/s0219498811004835.

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Let p be a prime number. Let G = Γ(R) be a ring graph, i.e. the zero-divisor graph of a commutative ring R. For an induced subgraph H of G, let CG(H) = {z ∈ V(G) ∣N(z) = V(H)}. Assume that in the graph G there exists an induced subgraph H which is isomorphic to the complete graph Kp-1, a vertex c ∈ CG(H), and a vertex z such that d(c, z) = 3. In this paper, we characterize the finite commutative rings R whose graphs G = Γ(R) have this property (called condition (Kp)).

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Chen, Xue-Gang, und Shinya Fujita. „On diameter and inverse degree of chemical graphs“. Applicable Analysis and Discrete Mathematics 7, Nr. 1 (2013): 83–93. http://dx.doi.org/10.2298/aadm121129022c.

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The inverse degree r(G) of a finite graph G = (V,E) is defined as r(G) = ?v?V 1/d(v), where d(v) is the degree of vertex v. In Discrete Math. 310 (2010), 940-946, Mukwembi posed the following conjecture: Let G be a connected chemical graph with diameter diam(G) and inverse degree r(G). Then diam(G) ? 12/5 r(G) + O(1). In this paper, we settle the conjecture affirmatively.

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Ramane, Harishchandra S., Deepa V. Kitturmath und Kavita Bhajantri. „Transmission-reciprocal transmission index and coindex of graphs“. Acta Universitatis Sapientiae, Informatica 14, Nr. 1 (01.08.2022): 84–103. http://dx.doi.org/10.2478/ausi-2022-0006.

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Abstract The transmission of a vertex u in a connected graph G is defined as σ(u) = Σv∈V(G) d(u, v) and reciprocal transmission of a vertex u is defined as r s ( u ) = ∑ v ∈ V ( G ) 1 d ( u , v ) rs(u) = \sum\nolimits_{v \in V\left( G \right)} {{1 \over {d\left( {u,v} \right)}}} , where d(u, v) is the distance between vertex u and v in G. In this paper we define new distance based topological index of a connected graph G called transmission-reciprocal transmission index T R T ( G ) = ∑ u v ∈ E ( G ) ( σ ( u ) r s ( u ) + σ ( v ) r s ( v ) ) TRT\left( G \right) = \sum\nolimits_{uv \in E\left( G \right)} {\left( {{{\sigma \left( u \right)} \over {rs\left( u \right)}} + {{\sigma \left( v \right)} \over {rs\left( v \right)}}} \right)} and its coindex T R T ¯ ( G ) = ∑ u v ∉ E ( G ) ( σ ( u ) r s ( u ) + σ ( v ) r s ( v ) ) \overline {TRT} \left( G \right) = \sum\nolimits_{uv \notin E\left( G \right)} {\left( {{{\sigma \left( u \right)} \over {rs\left( u \right)}} + {{\sigma \left( v \right)} \over {rs\left( v \right)}}} \right)} , where E(G) is the edge set of a graph G and establish the relation between TRT(G) and T R T ¯ ( G ) \overline {TRT} \left( G \right) (G). Further compute this index for some standard class of graphs and obtain bounds for it.

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Ning, Wenjie, Kun Wang und Hassan Raza. „Bicyclic Graphs with the Second-Maximum and Third-Maximum Degree Resistance Distance“. Journal of Mathematics 2021 (10.11.2021): 1–11. http://dx.doi.org/10.1155/2021/8722383.

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Let G = V , E be a connected graph. The resistance distance between two vertices u and v in G , denoted by R G u , v , is the effective resistance between them if each edge of G is assumed to be a unit resistor. The degree resistance distance of G is defined as D R G = ∑ u , v ⊆ V G d G u + d G v R G u , v , where d G u is the degree of a vertex u in G and R G u , v is the resistance distance between u and v in G . A bicyclic graph is a connected graph G = V , E with E = V + 1 . This paper completely characterizes the graphs with the second-maximum and third-maximum degree resistance distance among all bicyclic graphs with n ≥ 6 vertices.

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Jauhari, Mohammad Nafie. „On the Relation of the Total Graph of a Ring and a Product of Graphs“. Jurnal Matematika MANTIK 8, Nr. 2 (31.12.2022): 99–104. http://dx.doi.org/10.15642/mantik.2022.8.2.99-104.

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The total graph of a ring R, denoted as T(Γ(R)), is defined to be a graph with vertex set V(T(Γ(R)))=R and two distinct vertices u,v∈V(T(Γ(R))) are adjacent if and only if u+v∈Z(R), where Z(R) is the zero divisor of R. The Cartesian product of two graphs G and H is a graph with the vertex set V(G×H)=V(G)×V(H) and two distinct vertices (u_1,v_1 ) and (u_2,v_2 ) are adjacent if and only if: 1) u_1=u_2 and v_1 v_2∈H; or 2) v_1=v_2 and u_1 u_2∈E(G). An isomorphism of graphs G dan H is a bijection ϕ:V(G)→V(H) such that u,v∈V(G) are adjacent if and only if f(u),f(v)∈V(H) are adjacent. This paper proved that T(Γ(Z_2p )) and P_2×K_p are isomorphic for every odd prime p.

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Ghanem, Manal, Hasan Al-Ezeh und Ala’a Dabbour. „Locating Chromatic Number of Powers of Paths and Cycles“. Symmetry 11, Nr. 3 (18.03.2019): 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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Agustin, Ika Hesti, D. Dafik und A. Y. Harsya. „On r-dynamic coloring of some graph operations“. Indonesian Journal of Combinatorics 1, Nr. 1 (10.10.2016): 22. http://dx.doi.org/10.19184/ijc.2016.1.1.3.

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Let $G$ be a simple, connected and undirected graph. Let $r,k$ be natural number. By a proper $k$-coloring of a graph $G$, we mean a map $ c : V (G) \rightarrow S$, where $|S| = k$, such that any two adjacent vertices receive different colors. An $r$-dynamic $k$-coloring is a proper $k$-coloring $c$ of $G$ such that $|c(N (v))| \geq min\{r, d(v)\}$ for each vertex $v$ in $V(G)$, where $N (v)$ is the neighborhood of $v$ and $c(S) = \{c(v) : v \in S\}$ for a vertex subset $S$ . The $r$-dynamic chromatic number, written as $\chi_r(G)$, is the minimum $k$ such that $G$ has an $r$-dynamic $k$-coloring. Note that the $1$-dynamic chromatic number of graph is equal to its chromatic number, denoted by $\chi(G)$, and the $2$-dynamic chromatic number of graph has been studied under the name a dynamic chromatic number, denoted by $\chi_d(G)$. By simple observation it is easy to see that $\chi_r(G)\le \chi_{r+1}(G)$, however $\chi_{r+1}(G)-\chi_r(G)$ can be arbitrarily large, for example $\chi(Petersen)=2, \chi_d(Petersen)=3$, but $\chi_3(Petersen)=10$. Thus, finding an exact values of $\chi_r(G)$ is significantly useful. In this paper, we will show some exact values of $\chi_r(G)$ when $G$ is an operation of special graphs.

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Alfarisi, Ridho, Yuqing Lin, Joe Ryan, Dafik Dafik und Ika Hesti Agustin. „A Note on Multiset Dimension and Local Multiset Dimension of Graphs“. Statistics, Optimization & Information Computing 8, Nr. 4 (24.09.2020): 890–901. http://dx.doi.org/10.19139/soic-2310-5070-727.

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All graphs in this paper are nontrivial and connected simple graphs. For a set W = {s1,s2,...,sk} of verticesof G, the multiset representation of a vertex v of G with respect to W is r(v|W) = {d(v,s1),d(v,s2),...,d(v,sk)} whered(v,si) is the distance between of v and si. If the representation r(v|W)̸= r(u|W) for every pair of vertices u,v of a graph G, the W is called the resolving set of G, and the cardinality of a minimum resolving set is called the multiset dimension, denoted by md(G). A set W is a local resolving set of G if r(v|W) ̸= r(u|W) for every pair of adjacent vertices u,v of a graph G. The cardinality of a minimum local resolving set W is called local multiset dimension, denoted by µl(G). In our paper, we discuss the relationship between the multiset dimension and local multiset dimension of graphs and establish bounds of local multiset dimension for some families of graph.

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Deepa, T., M. Venkatachalam und Dafik. „On r− dynamic coloring of the gear graph families“. Proyecciones (Antofagasta) 40, Nr. 1 (01.02.2021): 01–15. http://dx.doi.org/10.22199/issn.0717-6279-2021-01-0001.

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An r-dynamic coloring of a graph G is a proper coloring c of the vertices such that |c(N(v))| ≥ min {r, d(v)}, for each v ∈ V (G). The r-dynamic chromatic number of a graph G is the minimum k such that G has an r-dynamic coloring with k colors. In this paper, we obtain the r−dynamic chromatic number of the middle, central and line graphs of the gear graph.

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Mohanapriya, N., K. Kalaiselvi, V. Aparna, Dafik und I. H. Agustin. „On r-dynamic coloring of central vertex join of path, cycle with certain graphs“. Journal of Physics: Conference Series 2157, Nr. 1 (01.01.2022): 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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Liu, Qun. „Resistance distance and Kirchhoff index in generalized R-vertex and R-edge corona for graphs“. Filomat 33, Nr. 6 (2019): 1593–604. http://dx.doi.org/10.2298/fil1906593l.

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For a graph G, the graph R(G) of a graph G is the graph obtained by adding a new vertex for each edge of G and joining each new vertex to both end vertices of the corresponding edge. Let I(G) be the set of newly added vertices, i.e I(G) = V(R(G))\ V(G). The generalized R-vertex corona of G and Hi for i = 1, 2, ?,n, denoted by R(G) ?? ^n i=1 Hi, is the graph obtained from R(G) and Hi by joining the i-th vertex of V(G) to every vertex in Hi. The generalized R-edge corona of G and Hi for i = 1, 2, ?,m, denoted by R(G)?^m i=1 Hi, is the graph obtained from R(G) and Hi by joining the i-th vertex of I(G) to every vertex in Hi. In this paper, we derive closed-form formulas for resistance distance and Kirchhoff index of R(G) ?? ^n i=1 Hi and R(G) ? ^m i=1 Hi whenever G and Hi are arbitrary graphs. These results generalize the existing results.

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Wu, Helin, Yong Ren und Feng Hu. „Dynkin game under g-expectation in continuous time“. Arabian Journal of Mathematics 9, Nr. 2 (09.03.2020): 459–70. http://dx.doi.org/10.1007/s40065-020-00281-2.

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Abstract In this paper, we investigate some kind of Dynkin game under g-expectation induced by backward stochastic differential equation (short for BSDE). The lower and upper value functions $$\underline{V}_t=ess\sup \nolimits _{\tau \in {\mathcal {T}_t}} ess\inf \nolimits _{\sigma \in {\mathcal {T}_t}}\mathcal {E}^g_t[R(\tau ,\sigma )]$$ V ̲ t = e s s sup τ ∈ T t e s s inf σ ∈ T t E t g [ R ( τ , σ ) ] and $$\overline{V}_t=ess\inf \nolimits _{\sigma \in {\mathcal {T}_t}} ess\sup \nolimits _{\tau \in {\mathcal {T}_t}}\mathcal {E}^g_t[R(\tau ,\sigma )]$$ V ¯ t = e s s inf σ ∈ T t e s s sup τ ∈ T t E t g [ R ( τ , σ ) ] are defined, respectively. Under some suitable assumptions, a pair of saddle points is obtained and the value function of Dynkin game $$V(t)=\underline{V}_t=\overline{V}_t$$ V ( t ) = V ̲ t = V ¯ t follows. Furthermore, we also consider the constrained case of Dynkin game.

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DAI, Guowei. „Degree sum and restricted {P2,P5}-factor in graphs“. Proceedings of the Romanian Academy, Series A: Mathematics, Physics, Technical Sciences, Information Science 24, Nr. 2 (28.06.2023): 105–11. http://dx.doi.org/10.59277/pra-ser.a.24.2.01.

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"For a graph $G$, a spanning subgraph $F$ of $G$ is called a $\{P_2,P_5\}$-factor if every component of $F$ is isomorphic to $P_2$ or $P_5$, where $P_i$ denotes the path of order $i$. A graph $G$ is called a $(\{P_2,P_5\},k)$-factor critical graph if $G-V'$ contains a $\{P_2,P_5\}$-factor for any $V'\subseteq V(G)$ with $|V'|=k$. A graph $G$ is called a $(\{P_2,P_5\},m)$-factor deleted graph if $G-E'$ has a $\{P_2,P_5\}$-factor for any $E'\subseteq E(G)$ with $|E'|=m$. The degree sum of $G$ is defined by $$\sigma_{r+1}(G)=\min_{X\subseteq V(G)}\Big\{\sum_{x\in X}d_G(x): X~\mathrm{is~an~independent~set~of}~r+1~\mathrm{vertices}\Big\}.$$ In this paper, using degree sum conditions, we demonstrate that (i) $G$ is a $(\{P_2,P_5\},k)$-factor critical graph if $\sigma_{r+1}(G)>\frac{(3n+4k-2)(r+1)}{7}$ and $\kappa(G)\geq k+r$; (ii) $G$ is a $(\{P_2,P_5\},m)$-factor deleted graph if $\sigma_{r+1}(G)>\frac{(3n+2m-2)(r+1)}{7}$ and $\kappa(G)\geq\frac{5m}{4}+r$."

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Raja, Rameez, S. Pirzada und Shane Redmond. „On locating numbers and codes of zero divisor graphs associated with commutative rings“. Journal of Algebra and Its Applications 15, Nr. 01 (07.09.2015): 1650014. http://dx.doi.org/10.1142/s0219498816500146.

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Let R be a commutative ring with identity and let G(V, E) be a graph. The locating number of the graph G(V, E) denoted by loc (G) is the cardinality of the minimal locating set W ⊆ V(G). To get the loc (G), we assign locating codes to the vertices V(G)∖W of G in such a way that every two vertices get different codes. In this paper, we consider the ratio of loc (G) to |V(G)| and show that there is a finite connected graph G with loc (G)/|V(G)| = m/n, where m < n are positive integers. We examine two equivalence relations on the vertices of Γ(R) and the relationship between locating sets and the cut vertices of Γ(R). Further, we obtain bounds for the locating number in zero-divisor graphs of a commutative ring and discuss the relation between locating number, domination number, clique number and chromatic number of Γ(R). We also investigate the locating number in Γ(R) when R is a finite product of rings.

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Ning, Wantao, und Hao Li. „The Generalized 3-Connectivity of Exchanged Folded Hypercubes“. Axioms 13, Nr. 3 (14.03.2024): 194. http://dx.doi.org/10.3390/axioms13030194.

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For S⊆V(G),κG(S) denotes the maximum number k of edge disjoint trees T1,T2,…,Tk in G, such that V(Ti)∩V(Tj)=S for any i,j∈{1,2,…,k} and i≠j. For an integer 2≤r≤|V(G)|, the generalized r-connectivity of G is defined as κr(G)=min{κG(S)|S⊆V(G)and|S|=r}. In fact, κ2(G) is the traditional connectivity of G. Hence, the generalized r-connectivity is an extension of traditional connectivity. The exchanged folded hypercube EFH(s,t), in which s≥1 and t≥1 are positive integers, is a variant of the hypercube. In this paper, we find that κ3(EFH(s,t))=s+1 with 3≤s≤t.

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Bian, Qiuju, und Sizhong Zhou. „Independence number, connectivity and fractional (g,f)-factors in graphs“. Filomat 29, Nr. 4 (2015): 757–61. http://dx.doi.org/10.2298/fil1504757b.

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Let G be a graph, and let g and f be two integer-valued functions defined on V(G) satisfying a ? 1(x) ? f (x)-r ? b - r for any x ? V(G), where a, b and r be three nonnegative integers with 1 ? a ? b - r. In this paper, we verify that G contains a fractional (g,f)-factor if its connectivity k(G) and independence number ?(G) satisfy k(G) ? max ((b+1)(b-r + 1)/2, (b-r + 1)2?(G)/4(a + r)). The result is best possible in some sense.

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Hu, Die, Peng Jin und Xianhua Tang. „The existence results for a class of generalized quasilinear Schrödinger equation with nonlocal term“. Electronic Research Archive 30, Nr. 5 (2022): 1973–98. http://dx.doi.org/10.3934/era.2022100.

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<abstract><p>In this paper, we discuss the generalized quasilinear Schrödinger equation with nonlocal term:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document} $ \begin{align} -\mathrm{div}(g^{2}(u)\nabla u)+g(u)g'(u)|\nabla u|^{2}+V(x)u = \left(|x|^{-\mu}\ast F(u)\right)f( u),\; \; x\in \mathbb{R}^{N}, \;\;\;\;\;\;\;\;({{\rm{P}}})\end{align} $ \end{document} </tex-math> </disp-formula></p> <p>where $ N\geq 3 $, $ \mu\in(0, N) $, $ g\in \mathbb{C}^{1}(\mathbb{R}, \mathbb{R}^{+}) $, $ V\in \mathbb{C}^{1}(\mathbb{R}^N, \mathbb{R}) $ and $ f\in \mathbb{C}(\mathbb{R}, \mathbb{R}) $. Under some "Berestycki-Lions type conditions" on the nonlinearity $ f $ which are almost necessary, we prove that problem $ (\rm P) $ has a nontrivial solution $ \bar{u}\in H^{1}(\mathbb{R}^{N}) $ such that $ \bar{v} = G(\bar{u}) $ is a ground state solution of the following problem</p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document} $ \begin{align} - \Delta v+V(x)\frac{G^{-1}(v)}{g(G^{-1}(v))} = \left(|x|^{-\mu}\ast F(G^{-1}(v))\right)f( G^{-1}(v)),\; \; x\in \mathbb{R}^{N}, \;\;\;\;\;\;\;\;({{\rm{\bar P}}})\end{align} $ \end{document} </tex-math> </disp-formula></p> <p>where $ G(t): = \int_{0}^{t} g(s) ds $. We also give a minimax characterization for the ground state solution $ \bar{v} $.</p></abstract>

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Madhavi Duggaraju, Radha, und Lipika Mazumdar. „Module Basis for Generalized Spline Modules“. Journal of the Indian Mathematical Society 89, Nr. 1-2 (27.01.2022): 32. http://dx.doi.org/10.18311/jims/2022/29295.

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Let G = (V,E) be a graph of order n. Let R be a commutative ring and I denote the set of all ideals of R. Let ? : E ? I be an edge labeling. A generalized spline of (G, ?) is a vertex labeling F : V ? R such that for each edge uv, F(u) ? F(v) ? ?(uv). The set R(G,) of all generalized splines of (G, ?) is an R-module. In this paper we determine conditions for a subset of R<sub>(G,?)</sub> to form a basis of R<sub>(G,?)</sub> for some classes of graphs.

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Ali, Akbar, Waqas Iqbal, Zahid Raza, Ekram E. Ali, Jia-Bao Liu, Farooq Ahmad und Qasim Ali Chaudhry. „Some Vertex/Edge-Degree-Based Topological Indices of r -Apex Trees“. Journal of Mathematics 2021 (21.10.2021): 1–8. http://dx.doi.org/10.1155/2021/4349074.

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In chemical graph theory, graph invariants are usually referred to as topological indices. For a graph G , its vertex-degree-based topological indices of the form BID G = ∑ u v ∈ E G β d u , d v are known as bond incident degree indices, where E G is the edge set of G , d w denotes degree of an arbitrary vertex w of G , and β is a real-valued-symmetric function. Those BID indices for which β can be rewritten as a function of d u + d v − 2 (that is degree of the edge u v ) are known as edge-degree-based BID indices. A connected graph G is said to be r -apex tree if r is the smallest nonnegative integer for which there is a subset R of V G such that R = r and G − R is a tree. In this paper, we address the problem of determining graphs attaining the maximum or minimum value of an arbitrary BID index from the class of all r -apex trees of order n , where r and n are fixed integers satisfying the inequalities n − r ≥ 2 and r ≥ 1 .

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Chen, Wenjing, und Fang Yu. „Schrödinger–Hardy system without the Ambrosetti–Rabinowitz condition on Carnot groups“. Electronic Journal of Qualitative Theory of Differential Equations, Nr. 23 (2024): 1–21. http://dx.doi.org/10.14232/ejqtde.2024.1.23.

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In this paper, we study the following Schrödinger–Hardy system { − Δ G u − μ ψ 2 r ( ξ ) 2 u = F u ( ξ , u , v ) i n Ω , − Δ G v − ν ψ 2 r ( ξ ) 2 v = F v ( ξ , u , v ) i n Ω , u = v = 0 o n ∂ Ω , where Ω is a smooth bounded domain on Carnot groups G , whose homogeneous dimension is Q ≥ 3 , Δ G denotes the sub-Laplacian operator on G , μ and ν are real parameters, r ( ξ ) is the natural gauge associated with fundamental solution of − Δ G on G , ψ is the geometrical function defined as ψ = | ∇ G r | , and ∇ G is the horizontal gradient associated with Δ G . The difficulty is not only the nonlinearities F u and F v without Ambrosetti–Rabinowitz condition, but also the Hardy terms and the structure on Carnot groups. We obtain the existence of nonnegative solution for this system by mountain pass theorem in a new framework.

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Tang, Zikai, und Hanyuan Deng. „Degree Kirchhoff Index of Bicyclic Graphs“. Canadian Mathematical Bulletin 60, Nr. 1 (01.03.2017): 197–205. http://dx.doi.org/10.4153/cmb-2016-063-5.

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AbstractLet G be a connected graph with vertex set V(G).The degree Kirchhoò index of G is defined as S'(G) = Σ{u,v}⊆V(G) d(u)d(v)R(u, v), where d(u) is the degree of vertex u, and R(u, v) denotes the resistance distance between vertices u and v. In this paper, we characterize the graphs having maximum and minimum degree Kirchhoò index among all n-vertex bicyclic graphs with exactly two cycles.

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F, Angdini Putri, Lyra Yulianti und Budi Rudianto. „DIMENSI METRIK DARI GRAF Amal(T rn, v)m untuk n = 5 dan m = 3“. Jurnal Matematika UNAND 8, Nr. 4 (13.12.2019): 77. http://dx.doi.org/10.25077/jmu.8.4.77-84.2019.

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Misalkan terdapat graf terhubung G = (V, E) dan himpunan terurut W ⊂ V (G), dengan W = {w1, w2, . . . , wk}, serta terdapat titik v ∈ V (G). Representasi titik v terhadap W yang dinotasikan dengan r(v|W), adalah k-vektorr(v|W) = (d(v, w1), d(v, w2), . . . , d(v, wk)).Jika untuk setiap dua titik u dan v di G diperoleh bahwa r(u|W) 6= r(v|W), maka W disebut sebagai himpunan pemisah (resolving set) untuk graf G. Kardinalitas dari himpunan pemisah minimum dinamakan dimensi metrik dari graf G yang dinotasikan dim(G). Graf amalgamasi tangga segitiga diperumum homogen adalah graf yang diperoleh dari hasil amalgamasi graf tangga segitiga diperumum yang sama untuk masing-masing graf. Graf tangga segitiga diperumum dinotasikan dengan T rn, untuk n ≥ 2. Pada paper ini dibahas tentang dimensi metrik dari graf Amal(T rn, v)m untuk n = 5 dan m = 3.Kata Kunci: Dimensi metrik, Himpunan pemisah, Graf Amal(T rn, v)m

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23

Zec, Tatjana. „On the Roman domination problem of some Johnson graphs“. Filomat 37, Nr. 7 (2023): 2067–75. http://dx.doi.org/10.2298/fil2307067z.

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A Roman domination function (RDF) on a graph G with a set of vertices V = V(G) is a function f : V ? {0, 1, 2} which satisfies the condition that each vertex v ? V such that f (v) = 0 is adjacent to at least one vertex u such that f (u) = 2. The minimum weight value of an RDF on graph G is called the Roman domination number (RDN) of G and it is denoted by ?R(G). An RDF for which ?R(G) is achieved is called a ?R(G)-function. This paper considers Roman domination problem for Johnson graphs Jn,2 and Jn,3. For Jn,2, n ? 4 it is proved that ?R(Jn,2) = n ? 1. New lower and upper bounds for Jn,3, n ? 6 are derived using results on the minimal coverings of pairs by triples. These bounds quadratically depend on dimension n.

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Almulhim, Ahlam, Abolape Deborah Akwu und Bana Al Subaiei. „The Perfect Roman Domination Number of the Cartesian Product of Some Graphs“. Journal of Mathematics 2022 (18.10.2022): 1–6. http://dx.doi.org/10.1155/2022/1957027.

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A perfect Roman dominating function on a graph G is a function f : V G ⟶ 0,1,2 for which every vertex v with f v = 0 is adjacent to exactly one neighbor u with f u = 2 . The weight of f is the sum of the weights of the vertices. The perfect Roman domination number of a graph G , denoted by γ R p G , is the minimum weight of a perfect Roman dominating function on G . In this paper, we prove that if G is the Cartesian product of a path P r and a path P s , a path P r and a cycle C s , or a cycle C r and a cycle C s , where r , s > 5 , then γ R p G ≤ 2 / 3 G .

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Koam, Ali N. A., Adnan Khalil, Ali Ahmad und Muhammad Azeem. „Cardinality bounds on subsets in the partition resolving set for complex convex polytope-like graph“. AIMS Mathematics 9, Nr. 4 (2024): 10078–94. http://dx.doi.org/10.3934/math.2024493.

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<abstract><p>Let $ G = (V, E) $ be a simple, connected graph with vertex set $ V(G) $ and $ E(G) $ edge set of $ G $. For two vertices $ a $ and $ b $ in a graph $ G $, the distance $ d(a, b) $ from $ a $ to $ b $ is the length of shortest path $ a-b $ path in $ G $. A $ k $-ordered partition of vertices of $ G $ is represented as $ {R}{p} = \{{R}{p_1}, {R}{p_2}, \dots, {R}{p_k}\} $ and the representation $ r(a|{R}{p}) $ of a vertex $ a $ with respect to $ {R}{p} $ is the vector $ (d(a|{R}{p_1}), d(a|{R}{p_2}), \dots, d(a|{R}{p_k})) $. The partition is called a resolving partition of $ G $ if $ r(a|{R}{p}) \ne r(b|{R}{p}) $ for all distinct $ a, b\in V(G) $. The partition dimension of a graph, denoted by $ pd(G) $, is the cardinality of a minimum resolving partition of $ G $. Computing precise and constant values for the partition dimension poses a interesting problem; therefore, it is possible to compute an upper bound for the partition dimension within a general family of graphs. In this paper, we studied partition dimension of the some families of convex polytopes, specifically $ \mathbb{T}_n $, $ \mathbb{U}_n $, $ \mathbb{V}_n $, and $ \mathbb{A}_n $, and proved that these graphs have constant partition dimension.</p></abstract>

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Ali, Ahmed Mohammed, und Asmaa Salah Aziz. „A Relation between D-Index and Wiener Index for r-Regular Graphs“. International Journal of Mathematics and Mathematical Sciences 2020 (22.02.2020): 1–6. http://dx.doi.org/10.1155/2020/6937863.

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For any two distinct vertices u and v in a connected graph G, let lPu,v=lP be the length of u−v path P and the D–distance between u and v of G is defined as: dDu,v=minplP+∑∀y∈VPdeg y, where the minimum is taken over all u−v paths P and the sum is taken over all vertices of u−v path P. The D-index of G is defined as WDG=1/2∑∀v,u∈VGdDu,v. In this paper, we found a general formula that links the Wiener index with D-index of a regular graph G. Moreover, we obtained different formulas of many special irregular graphs.

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Luo, Zuwen, und Liqiong Xu. „A Kind Of Conditional Vertex Connectivity Of Cayley Graphs Generated By Wheel Graphs“. Computer Journal 63, Nr. 9 (13.11.2019): 1372–84. http://dx.doi.org/10.1093/comjnl/bxz077.

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Abstract Let $G=(V(G), E(G))$ be a connected graph. A subset $T \subseteq V(G)$ is called an $R^{k}$-vertex-cut, if $G-T$ is disconnected and each vertex in $V(G)-T$ has at least $k$ neighbors in $G-T$. The cardinality of a minimum $R^{k}$-vertex-cut is the $R^{k}$-vertex-connectivity of $G$ and is denoted by $\kappa ^{k}(G)$. $R^{k}$-vertex-connectivity is a new measure to study the fault tolerance of network structures beyond connectivity. In this paper, we study $R^{1}$-vertex-connectivity and $R^{2}$-vertex-connectivity of Cayley graphs generated by wheel graphs, which are denoted by $AW_{n}$, and show that $\kappa ^{1}(AW_{n})=4n-7$ for $n\geq 6$; $\kappa ^{2}(AW_{n})=6n-12$ for $n\geq 6$.

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Brown, K. A., H. Marubayashi und P. F. Smith. „Group rings which are v-HC orders and Krull orders“. Proceedings of the Edinburgh Mathematical Society 34, Nr. 2 (Juni 1991): 217–28. http://dx.doi.org/10.1017/s0013091500007124.

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Let R be a ring and G a polycyclic-by-finite group. In this paper, it is determined, in terms of properties of R and G, when the group ring R[G] is a prime Krull order and when it is a price v-HC order. The key ingredient in obtaining both characterizations is the first author's earlier study of height one prime ideals in the ring R[G[.

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Wei, Xiaosha. „Directed Path 3-Arc-Connectivity of Cartesian Product Digraphs“. Symmetry 16, Nr. 4 (19.04.2024): 497. http://dx.doi.org/10.3390/sym16040497.

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Let D=(V(D),A(D)) be a digraph of order n and let r∈S⊆V(D) with 2≤|S|≤n. A directed (S,r)-Steiner path (or an (S,r)-path for short) is a directed path P beginning at r such that S⊆V(P). Arc-disjoint between two (S,r)-paths is characterized by the absence of common arcs. Let λS,rp(D) be the maximum number of arc-disjoint (S,r)-paths in D. The directed path k-arc-connectivity of D is defined as λkp(D)=min{λS,rp(D)∣S⊆V(D),S=k,r∈S}. In this paper, we shall investigate the directed path 3-arc-connectivity of Cartesian product λ3p(G□H) and prove that if G and H are two digraphs such that δ0(G)≥4, δ0(H)≥4, and κ(G)≥2, κ(H)≥2, then λ3p(G□H)≥min2κ(G),2κ(H); moreover, this bound is sharp. We also obtain exact values for λ3p(G□H) for some digraph classes G and H, and most of these digraphs are symmetric.

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Were, Hezron Saka, und Maurice Owino Oduor. „Classification of Unit Groups of Five Radical Zero Completely Primary Finite Rings Whose First and Second Galois Ring Module Generators Are of the Order p k , k = 2 , 3 , 4“. Journal of Mathematics 2022 (19.09.2022): 1–11. http://dx.doi.org/10.1155/2022/7867431.

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Let R 0 = G R p k r , p k be a Galois maximal subring of R so that R = R 0 ⊕ U ⊕ V ⊕ W ⊕ Y , where U , V , W , and Y are R 0 / p R 0 spaces considered as R 0 -modules, generated by the sets u 1 , ⋯ , u e , v 1 , ⋯ , v f , w 1 , ⋯ , w g , and y 1 , ⋯ , y h , respectively. Then, R is a completely primary finite ring with a Jacobson radical Z R such that Z R 5 = 0 and Z R 4 ≠ 0 . The residue field R / Z R is a finite field G F p r for some prime p and positive integer r . The characteristic of R is p k , where k is an integer such that 1 ≤ k ≤ 5 . In this paper, we study the structures of the unit groups of a commutative completely primary finite ring R with p ψ u i = 0 , ψ = 2 , 3 , 4 ; p ζ v j = 0 , ζ = 2 , 3 ; p w k = 0 , and p y l = 0 ; 1 ≤ i ≤ e , 1 ≤ j ≤ f , 1 ≤ k ≤ g , and 1 ≤ l ≤ h .

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31

Barnes, Benedict, E. D. J. Owusu-Ansah, S. K. Amponsah und C. Sebil. „The Proofs of Product Inequalities in Vector Spaces“. European Journal of Pure and Applied Mathematics 11, Nr. 2 (27.04.2018): 375–89. http://dx.doi.org/10.29020/nybg.ejpam.v11i2.3209.

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In this paper, we introduce the proofs of product inequalities:u v ≤ u + v , for all u, v ∈ [0, 2], and u + v ≤ u v , for allu, v ∈ [2, ∞). The first product inequality u v ≤ u + v holds forany two vectors in the interval [0, 1] in Holder’s space and also valid anytwo vectors in the interval [1, 2] in the Euclidean space. On the otherhand, the second product inequality u + v ≤ u v ∀u, v ∈ [2, ∞)only in Euclidean space. By applying the first product inequality to theL p spaces, we observed that if f : Ω → [0, 1], and g : Ω → R, thenf p g p ≤ f p + g p . Also, if f, g : Ω → R, then f p + g p ≤f p g p .

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Febrianti, Fifi, Lyra Yulianti und Narwen Narwen. „DIMENSI METRIK PADA GRAF AMALGAMASI TANGGA SEGITIGA DIPERUMUM HOMOGEN“. Jurnal Matematika UNAND 8, Nr. 1 (05.07.2019): 84. http://dx.doi.org/10.25077/jmu.8.1.84-90.2019.

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Misalkan terdapat G = (V, E) suatu graf terhubung dan misal terdapat dua titik u, v ∈ V . Jarak antara u dan v didefinisikan sebagai panjang lintasan terpendek antara u dan v yang dinotasikan dengan d(u, v). Misalkan terdapat himpunan terurut W ⊂ V (G), dengan W = {w1, w2, · · · , wk}. Misal terdapat titik v ∈ V (G). Representasi titik v terhadap W, dinotasikan r(v|W), adalah k-vektorr(v|W) = (d(v, w1), d(v, w2), · · · , d(v, wk)).Jika untuk setiap dua titik u dan v di G diperoleh bahwa r(u|W) 6= r(v|W), maka W dinamakan sebagai himpunan pemisah (resolving set) untuk G. Himpunan pemisah yang mempunyai kardinalitas minimum dinamakan himpunan pemisah minimum (minimum resolving set). Banyaknya anggota dari himpunan pemisah minimum ini dinamakan dimensi metrik dari G, dinotasikan dim(G). Graf amalgamasi graf tangga segitiga diperumum homogen adalah graf yang diperoleh dari hasil amalgamasi m buah graf tangga segitiga diperumum yang homogen, lebih sederhana dinotasikan dengan Amal{T rn, v}m. Pada paper ini dibahas dimensi metrik dari Amal{T rn, v}m dengan n = 3,n = 4 dan m = 2kata kunci: Dimensi Metrik, Himpunan pemisah,Representasi, Graf amalgamasi tangga segitiga diperumum homogen.Diterima: Direvisi:Dipublikasikan : KataKunci: Dimensi Metrik, Himpunan pemisah,Representasi, Graf amalgamsi tangga segitiga diperumum homogen.

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Lu, Jian, Shu-Bo Chen, Jia-Bao Liu, Xiang-Feng Pan und Ying-Jie Ji. „Further Results on the Resistance-Harary Index of Unicyclic Graphs“. Mathematics 7, Nr. 2 (20.02.2019): 201. http://dx.doi.org/10.3390/math7020201.

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The Resistance-Harary index of a connected graph G is defined as R H ( G ) = ∑ { u , v } ⊆ V ( G ) 1 r ( u , v ) , where r ( u , v ) is the resistance distance between vertices u and v in G. A graph G is called a unicyclic graph if it contains exactly one cycle and a fully loaded unicyclic graph is a unicyclic graph that no vertex with degree less than three in its unique cycle. Let U ( n ) and U ( n ) be the set of unicyclic graphs and fully loaded unicyclic graphs of order n, respectively. In this paper, we determine the graphs of U ( n ) with second-largest Resistance-Harary index and determine the graphs of U ( n ) with largest Resistance-Harary index.

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Rana, Akul, Anita Pal und Madhumangal Pal. „An Efficient Algorithm to Solve the Conditional Covering Problem on Trapezoid Graphs“. ISRN Discrete Mathematics 2011 (17.11.2011): 1–10. http://dx.doi.org/10.5402/2011/213084.

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Let G=(V,E) be a simple connected undirected graph. Each vertex v∈V has a cost c(v) and provides a positive coverage radius R(v). A distance duv is associated with each edge {u,v}∈E, and d(u,v) is the shortest distance between every pair of vertices u,v∈V. A vertex v can cover all vertices that lie within the distance R(v), except the vertex itself. The conditional covering problem is to minimize the sum of the costs required to cover all the vertices in G. This problem is NP-complete for general graphs, even it remains NP-complete for chordal graphs. In this paper, an O(n2) time algorithm to solve a special case of the problem in a trapezoid graph is proposed, where n is the number of vertices of the graph. In this special case, duv=1 for every edge {u,v}∈E, c(v)=c for every v∈V(G), and R(v)=R, an integer >1, for every v∈V(G). A new data structure on trapezoid graphs is used to solve the problem.

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Ghalavand, Ali, Ali Reza Ashrafi und Marzieh Pourbabaee. „Extremal Values of Randić Index among Some Classes of Graphs“. Mathematical Problems in Engineering 2021 (11.06.2021): 1–11. http://dx.doi.org/10.1155/2021/7758172.

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Suppose G is a simple graph with edge set E G . The Randić index R G is defined as R G = ∑ u v ∈ E G 1 / deg G u deg G v , where deg G u and deg G v denote the vertex degrees of u and v in G , respectively. In this paper, the first and second maximum of Randić index among all n − vertex c − cyclic graphs was computed. As a consequence, it is proved that the Randić index attains its maximum and second maximum on two classes of chemical graphs. Finally, we will present new lower and upper bounds for the Randić index of connected chemical graphs.

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Kusumawardani, I., Dafik, E. Y. Kurniawati, I. H. Agustin und R. Alfarisi. „On resolving efficient domination number of path and comb product of special graph“. Journal of Physics: Conference Series 2157, Nr. 1 (01.01.2022): 012012. http://dx.doi.org/10.1088/1742-6596/2157/1/012012.

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Abstract We use finite, connected, and undirected graph denoted by G. Let V (G) and E(G) be a vertex set and edge set respectively. A subset D of V (G) is an efficient dominating set of graph G if each vertex in G is either in D or adjoining to a vertex in D. A subset W of V (G) is a resolving set of G if any vertex in G is differently distinguished by its representation respect of every vertex in an ordered set W. Let W = {w 1, w 2, w 3, …, wk } be a subset of V (G). The representation of vertex υ ∈ G in respect of an ordered set W is r(υ|W) = (d(υ, w 1),d(υ, w 2), …, d(υ, wk )). The set W is called a resolving set of G if r(u|W) ≠ r(υ|W) ∀ u, υ ∈ G. A subset Z of V (G) is called the resolving efficient dominating set of graph G if it is an efficient dominating set and r(u|Z) ≠ r(υ|Z) ∀ u, υ ∈ G. Suppose γre (G) denotes the minimum cardinality of the resolving efficient dominating set. In other word we call a resolving efficient domination number of graphs. We obtained γreG of some comb product graphs in this paper, namely Pm ⊲ Pn , Sm ⊲ Pn , and Km ⊲ Pn .

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Raza, Hassan, Jia-Bao Liu, Muhammad Azeem und Muhammad Faisal Nadeem. „Partition Dimension of Generalized Petersen Graph“. Complexity 2021 (29.10.2021): 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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Firmansah, Fery, und Muhammad Ridlo Yuwono. „Odd Harmonious Labeling on Pleated of the Dutch Windmill Graphs“. CAUCHY 4, Nr. 4 (30.05.2017): 161. http://dx.doi.org/10.18860/ca.v4i4.4043.

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A graph G(p,q) with p=|V(G)| vertices and q=|E(G)| edges. The graph G(p,q) is said to be odd harmonious if there exist an injection f: V(G)->{0,1,2,...,2q-1} such that the induced function f*: E(G)->{1,2,3,...,2q-1} defined by f*(uv)=f(u)+f(v) which is a bijection and f is said to be odd harmonious labeling of G(p,q). In this paper we prove that pleated of the Dutch windmill graphs C_4^(k)(r) with k>=1 and r>=1 are odd harmonious graph. Moreover, we also give odd harmonious labeling construction for the union pleated of the Dutch windmill graph C_4^(k)(r) union C_4^(k)(r) with k>=1 and r>=1.

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Kalaiselvi, K., N. Mohanapriya und J. Vernold Vivin. „On r-dynamic coloring of comb graphs“. Notes on Number Theory and Discrete Mathematics 27, Nr. 2 (Juni 2021): 191–200. http://dx.doi.org/10.7546/nntdm.2021.27.2.191-200.

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An r-dynamic coloring of a graph G is a proper coloring of G such that every vertex in V(G) has neighbors in at least $\min\{d(v),r\}$ different color classes. The r-dynamic chromatic number of graph G denoted as $\chi_r (G)$, is the least k such that G has a coloring. In this paper we obtain the r-dynamic chromatic number of the central graph, middle graph, total graph, line graph, para-line graph and sub-division graph of the comb graph $P_n\odot K_1$ denoted by $C(P_n\odot K_1), M(P_n\odot K_1), T(P_n\odot K_1), L(P_n\odot K_1), P(P_n\odot K_1)$ and $S(P_n\odot K_1)$ respectively by finding the upper bound and lower bound for the r-dynamic chromatic number of the Comb graph.

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Bouchou, Ahmed, Mostafa Blidia und Mustapha Chellali. „Relations between the Roman k-domination and Roman domination numbers in graphs“. Discrete Mathematics, Algorithms and Applications 06, Nr. 03 (16.06.2014): 1450045. http://dx.doi.org/10.1142/s1793830914500451.

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Let G = (V, E) be a graph and let k be a positive integer. A Roman k-dominating function ( R k-DF) on G is a function f : V(G) → {0, 1, 2} such that every vertex u for which f(u) = 0 is adjacent to at least k vertices v1, v2, …, vk with f(vi) = 2 for i = 1, 2, …, k. The weight of an R k-DF is the value f(V(G)) = ∑u∈V(G) f(u) and the minimum weight of an R k-DF on G is called the Roman k-domination number γkR(G) of G. In this paper, we present relations between γkR(G) and γR(G). Moreover, we give characterizations of some classes of graphs attaining equality in these relations. Finally, we establish a relation between γkR(G) and γR(G) for {K1,3, K1,3+e}-free graphs and we characterize all such graphs G with γkR(G) = γR(G)+t, where [Formula: see text].

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ARUMUGAM, S., VARUGHESE MATHEW und JIAN SHEN. „ON FRACTIONAL METRIC DIMENSION OF GRAPHS“. Discrete Mathematics, Algorithms and Applications 05, Nr. 04 (Dezember 2013): 1350037. http://dx.doi.org/10.1142/s1793830913500377.

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A vertex x in a connected graph G = (V, E) is said to resolve a pair {u, v} of vertices of G if the distance from u to x is not equal to the distance from v to x. The resolving neighborhood for the pair {u, v} is defined as R{u, v} = {x ∈ V : d(u, x) ≠ d(v, x)}. A real valued function f : V → [0, 1] is a resolving function (RF) of G if f(R{u, v}) ≥ 1 for any two distinct vertices u, v ∈ V. The weight of f is defined by |f| = f(V) = ∑u∈Vf(v). The fractional metric dimension dim f(G) is defined by dim f(G) = min {|f| : f is a resolving function of G}. In this paper, we characterize graphs G for which [Formula: see text]. We also present several results on fractional metric dimension of Cartesian product of two connected graphs.

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Wu, Pu, Zepeng Li, Zehui Shao und Seyed Mahmoud Sheikholeslami. „Trees with equal Roman {2}-domination number and independent Roman {2}-domination number“. RAIRO - Operations Research 53, Nr. 2 (06.03.2019): 389–400. http://dx.doi.org/10.1051/ro/2018116.

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A Roman {2}-dominating function (R{2}DF) on a graph G =(V, E) is a function f : V → {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to either at least one vertex v with f(v) = 2 or two vertices v1, v2 with f(v1) = f(v2) = 1. The weight of an R{2}DF f is the value w(f) = ∑u∈Vf(u). The minimum weight of an R{2}DF on a graph G is called the Roman {2}-domination number γ{R2}(G) of G. An R{2}DF f is called an independent Roman {2}-dominating function (IR{2}DF) if the set of vertices with positive weight under f is independent. The minimum weight of an IR{2}DF on a graph G is called the independent Roman {2}-domination number i{R2}(G) of G. In this paper, we answer two questions posed by Rahmouni and Chellali.

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He, Fangguo, und Zhongxun Zhu. „The Extremal Cacti on Multiplicative Degree-Kirchhoff Index“. Mathematics 7, Nr. 1 (15.01.2019): 83. http://dx.doi.org/10.3390/math7010083.

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For a graph G, the resistance distance r G ( x , y ) is defined to be the effective resistance between vertices x and y, the multiplicative degree-Kirchhoff index R ∗ ( G ) = ∑ { x , y } ⊂ V ( G ) d G ( x ) d G ( y ) r G ( x , y ) , where d G ( x ) is the degree of vertex x, and V ( G ) denotes the vertex set of G. L. Feng et al. obtained the element in C a c t ( n ; t ) with first-minimum multiplicative degree-Kirchhoff index. In this paper, we first give some transformations on R ∗ ( G ) , and then, by these transformations, the second-minimum multiplicative degree-Kirchhoff index and the corresponding extremal graph are determined, respectively.

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Ma, Yuzheng, Yubin Gao und Yanling Shao. „New Bounds for the Generalized Distance Spectral Radius/Energy of Graphs“. Mathematical Problems in Engineering 2022 (09.11.2022): 1–9. http://dx.doi.org/10.1155/2022/9562730.

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Let G be a simple connected graph with vertex set V G = v 1 , v 2 , … , v n and d v i be the degree of the vertex v i . Let D G be the distance matrix and T r G be the diagonal matrix of the vertex transmissions of G . The generalized distance matrix of G is defined as D α G = α T r G + 1 − α D G , where 0 ≤ α ≤ 1 . If λ 1 , λ 2 , … , λ n are the eigenvalues of D α G , then the generalized distance spectral radius of G is defined as ρ D α G = max 1 ≤ i ≤ n λ i . The generalized distance energy of G is E D α G = ∑ i = 1 n | λ i − 2 α W G / n | , where W G is the Wiener index of G . In this paper, we give some bounds of the generalized distance spectral radius and the generalized distance energy.

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Ferme, Jasmina, und Dasa Stesl. „On distance dominator packing coloring in graphs“. Filomat 35, Nr. 12 (2021): 4005–16. http://dx.doi.org/10.2298/fil2112005f.

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Let G be a graph and let S = (s1,s2,..., sk) be a non-decreasing sequence of positive integers. An S-packing coloring of G is a mapping c : V(G) ? {1, 2,..., k} with the following property: if c(u) = c(v) = i, then d(u,v) > si for any i ? {1, 2,...,k}. In particular, if S = (1, 2, 3, ..., k), then S-packing coloring of G is well known under the name packing coloring. Next, let r be a positive integer and u,v ? V(G). A vertex u r-distance dominates a vertex v if dG(u, v)? r. In this paper, we present a new concept of a coloring, namely distance dominator packing coloring, defined as follows. A coloring c is a distance dominator packing coloring of G if it is a packing coloring of G and for each x ? V(G) there exists i ? {1,2, 3,...} such that x i-distance dominates each vertex from the color class of color i. The smallest integer k such that there exists a distance dominator packing coloring of G using k colors, is the distance dominator packing chromatic number of G, denoted by ?d?(G). In this paper, we provide some lower and upper bounds on the distance dominator packing chromatic number, characterize graphs G with ?d?(G) ? {2,3}, and provide the exact values of ?d?(G) when G is a complete graph, a star, a wheel, a cycle or a path. In addition, we consider the relation between ?? (G) and ?d?(G) for a graph G.

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Kang, Cong, Ismael Yero und Eunjeong Yi. „The fractional k-metric dimension of graphs“. Applicable Analysis and Discrete Mathematics 13, Nr. 1 (2019): 203–23. http://dx.doi.org/10.2298/aadm170712023k.

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Let G be a graph with vertex set V (G). For any two distinct vertices x and y of G, let R{x,y} denote the set of vertices z such that the distance from x to z is not equal to the distance from y to z in G. For a function g defined on V (G) and for U ? V (G), let g(U)= ? s?U g(s). Let k(G) = min{|R{x,y}|: x ? y and x,y ? V (G)}. For any real number k ?[1,k(G)], a real-valued function g : V(G)?[0,1] is a k-resolving function of G if g(R{x,y}) ? k for any two distinct vertices x,y ? V(G). The fractional k-metric dimension, dimkf(G), of G is min{g(V(G)):g is a k-resolving function of G}. In this paper, we initiate the study of the fractional k-metric dimension of graphs. For a connected graph G and k?[1,k(G)], it's easy to see that k ? dimkf(G)? k|V(G)|/k(G); we characterize graphs G satisfying dimkf(G)=k and dimkf(G)=|V(G)| respectively. We show that dimkf(G) ? k dimf(G) for any k ? [1,k(G)], and we give an example showing that dimkf(G)- k dimf(G) can be arbitrarily large for some k?(1,k(G)]; we also describe a condition for which dimkf(G) = kdimf(G) holds. We determine the fractional k-metric dimension for some classes of graphs, and conclude with two open problems, including whether ?(k) = dimkf(G) is a continuous function of k on every connected graph G.

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Mayora, Citra, Narwen Narwen und Des Welyyanti. „DIMENSI METRIK DARI GRAF SPINNER (C3 × P2) Kn UNTUK n = 1“. Jurnal Matematika UNAND 7, Nr. 4 (19.02.2019): 1. http://dx.doi.org/10.25077/jmu.7.4.1-6.2018.

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Misalkan u dan v adalah titik-titik dalam graf terhubung G. Jarak d(u, v) adalah panjang lintasan terpendek antara u dan v pada graf G. Bila diberikan himpunan terurut W = {w1, w2, w3, · · · , wk} dari titik-titik dalam graf terhubung G dan titik v ∈ V (G), representasi dari v terhadap W adalah k-vektor yang dapat ditulis dengan r(v|W) = (d(v, w1), d(v, w2), · · · , d(v, wk)). Jika r(v|W) untuk setiap titik v ∈ (G) berbeda, maka W disebut himpunan pembeda dari V (G). Himpunan pembeda dengan kardinalitas minimum disebut himpunan pembeda minimum dan kardinalitas dari basis metrik tersebut dinamakan dimensi metrik dari graf G dan dinotasikan dengan dim(G). Graf spinner adalah perkalian kartesius antara graf C3 dan graf P2 yang menghasilkan graf C3 × P2, kemudian graf C3 × P2 tersebut dikoronakan dengan graf komplemen Kn yaitu Kn, sehingga graf spinner tersebut dapat dinotasikan dengan (C3 ×P2)Kn. Pada paper ini akan dibahas dimensi metrik dari graf spinner (C3 × P2) Kn untuk n = 1.Kata Kunci: Dimensi metrik, Himpunan pembeda, Representasi, Hasilkali kartesius, Graf korona

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Zhou, Sizhong, Lan Xu und Yang Xu. „A sufficient condition for the existence of a k-factor excluding a given r-factor“. Applied Mathematics and Nonlinear Sciences 2, Nr. 1 (10.01.2017): 13–20. http://dx.doi.org/10.21042/amns.2017.1.00002.

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AbstractLet G be a graph, and let k, r be nonnegative integers with k ≥ 2. A k-factor of G is a spanning subgraph F of G such that dF(x) = k for each x ∈ V (G), where dF(x) denotes the degree of x in F. For S ⊆ V (G), NG(S) = ∪x∊SNG(x). The binding number of G is defined by bind$\begin{array}{} (G) = {\rm{min }}\{ \frac{{|{N_G}(S)|}}{{|S|}}:\emptyset \ne S \subset V(G),{N_G}(S) \ne V(G)\} \end{array}$. In this paper, we obtain a binding number and neighborhood condition for a graph to have a k-factor excluding a given r-factor. This result is an extension of the previous results.

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Raza, Mohsin, Dalal Awadh Alrowaili, Muhammad Javaid und Khurram Shabbir. „Computing Bounds of Fractional Metric Dimension of Metal Organic Graphs“. Journal of Chemistry 2021 (12.03.2021): 1–12. http://dx.doi.org/10.1155/2021/5539569.

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Metal organic graphs are hollow structures of metal atoms that are connected by ligands, where metal atoms are represented by the vertices and ligands are referred as edges. A vertex x resolves the vertices u and v of a graph G if d u , x ≠ d v , x . For a pair u , v of vertices of G , R u , v = x ∈ V G : d x , u ≠ d x , v is called its resolving neighbourhood set. For each pair of vertices u and v in V G , if f R u , v ≥ 1 , then f from V G to the interval 0,1 is called resolving function. Moreover, for two functions f and g , f is called minimal if f ≤ g and f v ≠ g v for at least one v ∈ V G . The fractional metric dimension (FMD) of G is denoted by dim f G and defined as dim f G = min g : g is a minimal resolving function of G , where g = ∑ v ∈ V G g v . If we take a pair of vertices u , v of G as an edge e = u v of G , then it becomes local fractional metric dimension (LFMD) dim l f G . In this paper, local fractional and fractional metric dimensions of MOG n are computed for n ≅ 1 mod 2 in the terms of upper bounds. Moreover, it is obtained that metal organic is one of the graphs that has the same local and fractional metric dimension.

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Bhakta, Mousomi, Souptik Chakraborty, Olimpio H. Miyagaki und Patrizia Pucci. „Fractional elliptic systems with critical nonlinearities“. Nonlinearity 34, Nr. 11 (28.09.2021): 7540–73. http://dx.doi.org/10.1088/1361-6544/ac24e5.

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Abstract This paper deals with existence, uniqueness and multiplicity of positive solutions to the following nonlocal system of equations: }0\quad \text{in}\hspace{2pt}{\mathbb{R}}^{N},\end{aligned}\right.\qquad \qquad \qquad \qquad (\mathcal{S})\end{equation*}?> ( − Δ ) s u = α 2 s * | u | α − 2 u | v | β + f ( x ) in R N , ( − Δ ) s v = β 2 s * | v | β − 2 v | u | α + g ( x ) in R N , u , v > 0 in R N , ( S ) where 0 < s < 1, N > 2s, α, β > 1, α + β = 2N/(N − 2s), and f, g are nonnegative functionals in the dual space of H ˙ s ( R N ) , i.e., 〈 ( H ˙ s ) ′ f , u 〉 H ˙ s ⩾ 0 , whenever u is a nonnegative function in H ˙ s ( R N ) . When f = 0 = g, we show that the ground state solution of ( S ) is unique. On the other hand, when f and g are nontrivial nonnegative functionals with ker(f) = ker(g), then we establish the existence of at least two different positive solutions of ( S ) provided that ‖ f ‖ ( H ˙ s ) ′ and ‖ g ‖ ( H ˙ s ) ′ are small enough. Moreover, we also provide a global compactness result, which gives a complete description of the Palais–Smale sequences of the above system.

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