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Journal articles on the topic 'Graphs; Non-negative'

Author: Grafiati

Published: 4 June 2021

Last updated: 7 February 2022

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1

Teng, Wenshun, and Huijuan Wang. "Vertex arboricity of graphs embedded in a surface of non-negative Euler characteristic." Discrete Mathematics, Algorithms and Applications 12, no. 06 (July 30, 2020): 2050080. http://dx.doi.org/10.1142/s1793830920500809.

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The vertex arboricity [Formula: see text] of a graph [Formula: see text] is the minimum number of colors the vertices of the graph [Formula: see text] can be colored so that every color class induces an acyclic subgraph of [Formula: see text]. There are many results on the vertex arboricity of planar graphs. In this paper, we replace planar graphs with graphs which can be embedded in a surface [Formula: see text] of Euler characteristic [Formula: see text]. We prove that for the graph [Formula: see text] which can be embedded in a surface [Formula: see text] of Euler characteristic [Formula: see text] if no [Formula: see text]-cycle intersects a [Formula: see text]-cycle, or no [Formula: see text]-cycle intersects a [Formula: see text]-cycle, then [Formula: see text] in addition to the [Formula: see text]-regular quadrilateral mesh.

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2

YANHAONA, MUHAMMAD NUR, MD SHAMSUZZOHA BAYZID, and MD SAIDUR RAHMAN. "DISCOVERING PAIRWISE COMPATIBILITY GRAPHS." Discrete Mathematics, Algorithms and Applications 02, no. 04 (December 2010): 607–23. http://dx.doi.org/10.1142/s1793830910000917.

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Let T be an edge weighted tree, let dT(u, v) be the sum of the weights of the edges on the path from u to v in T, and let d min and d max be two non-negative real numbers such that d min ≤ d max . Then a pairwise compatibility graph of T for d min and d max is a graph G = (V, E), where each vertex u' ∈ V corresponds to a leaf u of T and there is an edge (u', v') ∈ E if and only if d min ≤ dT(u, v) ≤ d max . A graph G is called a pairwise compatibility graph (PCG) if there exists an edge weighted tree T and two non-negative real numbers d min and d max such that G is a pairwise compatibility graph of T for d min and d max . Kearney et al. conjectured that every graph is a PCG [3]. In this paper, we refute the conjecture by showing that not all graphs are PCG s . Moreover, we recognize several classes of graphs as pairwise compatibility graphs. We identify two restricted classes of bipartite graphs as PCG. We also show that the well known tree power graphs and some of their extensions are PCGs.

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3

Jiménez González, Jesús Arturo. "Incidence graphs and non-negative integral quadratic forms." Journal of Algebra 513 (November 2018): 208–45. http://dx.doi.org/10.1016/j.jalgebra.2018.07.020.

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4

Derikvand, Tajedin, and Mohammad Reza Oboudi. "Small graphs with exactly two non-negative eigenvalues." Algebraic structures and their applications 4, no. 1 (August 1, 2017): 1–18. http://dx.doi.org/10.29252/asta.4.1.1.

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5

Zhang, Kewei. "On non-negative quasiconvex functions with unbounded zero sets." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 127, no. 2 (1997): 411–22. http://dx.doi.org/10.1017/s0308210500023726.

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We construct nontrivial, non-negative quasiconvex functions denned on M2×2 with p-th order growth such that the zero sets of the functions are Lipschitz graphs of mappings from subsets of a fixed two-dimensional subspace to its orthogonal complement. We assume that the graphs do not have rank-one connections with the Lipschitz constants sufficiently small. In particular, we are able to construct quasiconvex functions which are homogeneous of degree p (p > 1) and ‘conjugating’ invariant.

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6

Oboudi, Mohammad Reza. "Characterization of graphs with exactly two non-negative eigenvalues." Ars Mathematica Contemporanea 12, no. 2 (December 23, 2016): 271–86. http://dx.doi.org/10.26493/1855-3974.1077.5b6.

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7

Koledin, Tamara, and Zoran Stanić. "Regular bipartite graphs with three distinct non-negative eigenvalues." Linear Algebra and its Applications 438, no. 8 (April 2013): 3336–49. http://dx.doi.org/10.1016/j.laa.2012.12.036.

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8

Chung, Fan, Yong Lin, and S. T. Yau. "Harnack inequalities for graphs with non-negative Ricci curvature." Journal of Mathematical Analysis and Applications 415, no. 1 (July 2014): 25–32. http://dx.doi.org/10.1016/j.jmaa.2014.01.044.

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9

Alomari, Omar, Mohammad Abudayah, and Torsten Sander. "The non-negative spectrum of a digraph." Open Mathematics 18, no. 1 (February 19, 2020): 22–35. http://dx.doi.org/10.1515/math-2020-0005.

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Abstract Given the adjacency matrix A of a digraph, the eigenvalues of the matrix AAT constitute the so-called non-negative spectrum of this digraph. We investigate the relation between the structure of digraphs and their non-negative spectra and associated eigenvectors. In particular, it turns out that the non-negative spectrum of a digraph can be derived from the traditional (adjacency) spectrum of certain undirected bipartite graphs.

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10

B. Boomadevi, V. Gopal, and B. Boomadevi. "ON SIGNED (NON-NEGATIVE) MAJORITY TOTAL DOMINATION OF SOME GRAPHS." Advances in Mathematics: Scientific Journal 9, no. 4 (July 3, 2020): 2039–45. http://dx.doi.org/10.37418/amsj.9.4.62.

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11

Mathai, Jincy P., Sudev Naduvath, and Satheesh Sreedharan. "On ideal sumset labelled graphs." Proyecciones (Antofagasta) 40, no. 2 (April 2021): 371–84. http://dx.doi.org/10.22199/issn.0717-6279-2021-02-0022.

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The sumset of two sets A and B of integers, denoted by A + B, is defined as A+B = {a+b : a ∈ A, b ∈ B}. Let X be a non-empty set of non-negative integers. A sumset labelling of a graph G is an injective function f : V (G) → P(X) − {∅} such that the induced function f+ : E(G) → P(X)−{∅} is defined by f+(uv) = f(u) +f(v) ∀uv ∈ E(G). In this paper, we introduce the notion of ideal sumset labelling of graph and discuss the admissibility of this labelling by certain graph classes and discuss some structural characterization of those graphs.

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12

Biró, Csaba, and Udayan B. Darji. "Generating Infinite Random Graphs." Proceedings of the Edinburgh Mathematical Society 61, no. 3 (May 22, 2018): 847–68. http://dx.doi.org/10.1017/s0013091517000487.

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AbstractWe define a growing model of random graphs. Given a sequence of non-negative integers {dn}n=0∞ with the property that di≤i, we construct a random graph on countably infinitely many vertices v0, v1… by the following process: vertex vi is connected to a subset of {v0, …, vi−1} of cardinality di chosen uniformly at random. We study the resulting probability space. In particular, we give a new characterization of random graphs, and we also give probabilistic methods for constructing infinite random trees.

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13

Zając, Katarzyna. "On the structure of loop-free non-negative edge-bipartite graphs." Linear Algebra and its Applications 579 (October 2019): 262–83. http://dx.doi.org/10.1016/j.laa.2019.06.002.

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14

Cardoso, Domingos, Paula Carvalho, Paula Rama, Slobodan Simic, and Zoran Stanic. "Lexicographic polynomials of graphs and their spectra." Applicable Analysis and Discrete Mathematics 11, no. 2 (2017): 258–72. http://dx.doi.org/10.2298/aadm1702258c.

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For a (simple) graph H and non-negative integers c0, c1,..., cd (cd ? 0), p(H) = ?dk=0 ck?Hk is the lexicographic polynomial in H of degree d, where the sum of two graphs is their join and ck?Hk is the join of ck copies of Hk. The graph Hk is the kth power of H with respect to the lexicographic product (H0 = K1). The spectrum (if H is connected and regular) and the Laplacian spectrum (in general case) of p(H) are determined in terms of the spectrum of H and ck's. Constructions of infinite families of cospectral or integral graphs are announced.

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15

Sudev, N. K., K. P. Chithra, and K. A. Germina. "Switched signed graphs of integer additive set-valued signed graphs." Discrete Mathematics, Algorithms and Applications 09, no. 04 (August 2017): 1750043. http://dx.doi.org/10.1142/s1793830917500434.

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Let [Formula: see text] denote a set of non-negative integers and [Formula: see text] be its power set. An integer additive set-labeling (IASL) of a graph [Formula: see text] is an injective set-valued function [Formula: see text] such that the induced function [Formula: see text] is defined by [Formula: see text], where [Formula: see text] is the sumset of [Formula: see text] and [Formula: see text]. An IASL of a signed graph [Formula: see text] is an IASL of its underlying graph [Formula: see text] together with the signature [Formula: see text] defined by [Formula: see text]. A marking of a signed graph is an injective map [Formula: see text], defined by [Formula: see text] for all [Formula: see text]. Switching of signed graph is the process of changing the sign of the edges in [Formula: see text] whose end vertices have different signs. In this paper, we discuss certain characteristics of the switched signed graphs of certain types of integer additive set-labeled signed graphs.

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16

Carlson, W., A. Ford, E. Harris, J. Rosen, C. Tamon, and K. Wrobel. "Universal Mixing of Quantum Walk on Graphs." Quantum Information and Computation 7, no. 8 (November 2007): 738–51. http://dx.doi.org/10.26421/qic7.8-4.

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We study the set of probability distributions visited by a continuous-time quantum walk on graphs. An edge-weighted graph $G$ is {\em universal mixing} if the instantaneous or average probability distribution of the quantum walk on $G$ ranges over all probability distributions on the vertices as the weights are varied over non-negative reals. The graph is {\em uniform} mixing if it visits the uniform distribution. Our results include the following: 1) All weighted complete multipartite graphs are instantaneous universal mixing. This is in contrast to the fact that no {\em unweighted} complete multipartite graphs are uniform mixing (except for the four-cycle $K_{2,2}$). 2) For all $n \ge 1$, the weighted claw $K_{1,n}$ is a minimally connected instantaneous universal mixing graph. In fact, as a corollary, the unweighted $K_{1,n}$ is instantaneous uniform mixing. This adds a new family of uniform mixing graphs to a list that so far contains only the hypercubes. 3) Any weighted graph is average almost-uniform mixing unless its spectral type is sublinear in the size of the graph. This provides a nearly tight characterization for average uniform mixing on circulant graphs. 4) No weighted graphs are average universal mixing. This shows that weights do not help to achieve average universal mixing, unlike the instantaneous case. Our proofs exploit the spectra of the underlying weighted graphs and path collapsing arguments.

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17

CALAMONERI, TIZIANA, ROSSELLA PETRESCHI, and BLERINA SINAIMERI. "ON THE PAIRWISE COMPATIBILITY PROPERTY OF SOME SUPERCLASSES OF THRESHOLD GRAPHS." Discrete Mathematics, Algorithms and Applications 05, no. 02 (June 2013): 1360002. http://dx.doi.org/10.1142/s1793830913600021.

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A graph G is called a pairwise compatibility graph (PCG) if there exists a positive edge weighted tree T and two non-negative real numbers d min and d max such that each leaf lu of T corresponds to a node u ∈ V and there is an edge (u, v) ∈ E if and only if d min ≤ dT (lu, lv) ≤ d max , where dT (lu, lv) is the sum of the weights of the edges on the unique path from lu to lv in T. In this paper we study the relations between the pairwise compatibility property and superclasses of threshold graphs, i.e., graphs where the neighborhoods of any couple of nodes either coincide or are included one into the other. Namely, we prove that some of these superclasses belong to the PCG class. Moreover, we tackle the problem of characterizing the class of graphs that are PCGs of a star, deducing that also these graphs are a generalization of threshold graphs.

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18

Al-Harere, Manal N., and Mohammed A. Abdlhusein. "Pitchfork domination in graphs." Discrete Mathematics, Algorithms and Applications 12, no. 02 (March 9, 2020): 2050025. http://dx.doi.org/10.1142/s1793830920500251.

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In this paper, a new model of domination in graphs called the pitchfork domination is introduced. Let [Formula: see text] be a finite, simple and undirected graph without isolated vertices, a subset [Formula: see text] of [Formula: see text] is a pitchfork dominating set if every vertex [Formula: see text] dominates at least [Formula: see text] and at most [Formula: see text] vertices of [Formula: see text], where [Formula: see text] and [Formula: see text] are non-negative integers. The domination number of [Formula: see text], denotes [Formula: see text] is a minimum cardinality over all pitchfork dominating sets in [Formula: see text]. In this work, pitchfork domination when [Formula: see text] and [Formula: see text] is studied. Some bounds on [Formula: see text] related to the order, size, minimum degree, maximum degree of a graph and some properties are given. Pitchfork domination is determined for some known and new modified graphs. Finally, a question has been answered and discussed that; does every finite, simple and undirected graph [Formula: see text] without isolated vertices have a pitchfork domination or not?

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19

Ganesamurthy, S., P. Paulraja, and R. Srimathi. "Multidecompositions of line graphs of complete graphs." Discrete Mathematics, Algorithms and Applications 11, no. 03 (June 2019): 1950035. http://dx.doi.org/10.1142/s1793830919500356.

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By a [Formula: see text]-decomposition of a graph [Formula: see text] we mean a decomposition of [Formula: see text] into [Formula: see text] copies of [Formula: see text] [Formula: see text] copies of [Formula: see text] and [Formula: see text] copies of [Formula: see text], where [Formula: see text] are non-negative integers. In this paper, it is proved that the necessary conditions for the existence of a [Formula: see text]-decomposition of [Formula: see text] are sufficient, where B, the bowtie, is the graph with two triangles having exactly one common vertex. Further, it is shown that the line graph of [Formula: see text] [Formula: see text] has a [Formula: see text]-decomposition except for [Formula: see text].

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20

Ganie, Hilal A., S. Pirzada, Rezwan Ul Shaban, and X. Li. "Upper bounds for the sum of Laplacian eigenvalues of a graph and Brouwer’s conjecture." Discrete Mathematics, Algorithms and Applications 11, no. 02 (April 2019): 1950028. http://dx.doi.org/10.1142/s1793830919500289.

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Consider a simple graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] having Laplacian eigenvalues [Formula: see text]. Let [Formula: see text] be the sum of [Formula: see text] largest Laplacian eigenvalues of [Formula: see text]. Brouwer conjectured that [Formula: see text] for all [Formula: see text]. We obtain an upper bound for [Formula: see text] in terms of the clique number [Formula: see text], the number of vertices [Formula: see text] and the non-negative integers [Formula: see text] associated to the structure of the graph [Formula: see text]. We show that the Brouwer’s conjecture holds true for some new families of graphs. We use the same technique to prove that the Brouwer’s conjecture is true for a subclass of split graphs (It is already known that Brouwer’s conjecture holds for split graphs).

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21

Lepovic, Mirko. "On strongly regular graphs with m2 = qm3 and m3 = qm2 where q ∈ Q." Publications de l'Institut Math?matique (Belgrade) 109, no. 123 (2021): 35–60. http://dx.doi.org/10.2298/pim2123035l.

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We say that a regular graph G of order n and degree r ? 1 (which is not the complete graph) is strongly regular if there exist non-negative integers ? and ? such that |Si ? Sj | = ? for any two adjacent vertices i and j, and |Si ? Sj | = ? for any two distinct non-adjacent vertices i and j, where Sk denotes the neighborhood of the vertex k. Let ?1 = r, ?2 and ?3 be the distinct eigenvalues of a connected strongly regular graph. Let m1 = 1, m2 and m3 denote the multiplicity of r, ?2 and ?3, respectively. We here describe the parameters n, r, ? and ? for strongly regular graphs with m2 = qm3 and m3 = qm2 for q = 3/2, 4/3, 5/2, 5/3, 5/4, 6/5.

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22

LYONS, RUSSELL. "Identities and Inequalities for Tree Entropy." Combinatorics, Probability and Computing 19, no. 2 (December 15, 2009): 303–13. http://dx.doi.org/10.1017/s0963548309990605.

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The notion of tree entropy was introduced by the author as a normalized limit of the number of spanning trees in finite graphs, but is defined on random infinite rooted graphs. We give some new expressions for tree entropy; one uses Fuglede–Kadison determinants, while another uses effective resistance. We use the latter to prove that tree entropy respects stochastic domination. We also prove that tree entropy is non-negative in the unweighted case, a special case of which establishes Lück's Determinant Conjecture for Cayley-graph Laplacians. We use techniques from the theory of operators affiliated to von Neumann algebras.

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23

Sudev, N. K., and K. A. Germina. "A study on integer additive set-valuations of signed graphs." Carpathian Mathematical Publications 7, no. 2 (December 15, 2015): 236–46. http://dx.doi.org/10.15330/cmp.7.2.236-246.

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Let $\mathbb{N}_0$ denote the set of all non-negative integers and $\mathcal{P}(\mathbb{N}_0)$ be its power set. An integer additive set-labeling (IASL) of a graph $G$ is an injective set-valued function $f:V(G)\to\mathcal{P}(\mathbb{N}_0)\setminus\{\emptyset\}$ such that the induced function $f^+:E(G) \to \mathcal{P}(\mathbb{N}_0)\setminus \{\emptyset\}$ is defined by $f^+(uv) = f(u)+ f(v)$, where $f(u)+f(v)$ is the sumset of $f(u)$ and $f(v)$. A graph which has an IASL is usually called an IASL-graph. An IASL $f$ of a graph $G$ is said to be an integer additive set-indexer (IASI) of $G$ if the associated function $f^+$ is also injective. In this paper, we define the notion of integer additive set-labeling of signed graphs and discuss certain properties of signed graphs which admits certain types of integer additive set-labelings.

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24

Gąsiorek, Marcin, Daniel Simson, and Katarzyna Zając. "A Gram classification of non-negative corank-two loop-free edge-bipartite graphs." Linear Algebra and its Applications 500 (July 2016): 88–118. http://dx.doi.org/10.1016/j.laa.2016.03.007.

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25

Zając, Katarzyna. "On polynomial time inflation algorithm for loop-free non-negative edge-bipartite graphs." Discrete Applied Mathematics 283 (September 2020): 28–43. http://dx.doi.org/10.1016/j.dam.2019.12.002.

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26

Yousaf, Shamaila, Akhlaq Ahmad Bhatti, and Akbar Ali. "On the Minimum Variable Connectivity Index of Unicyclic Graphs with a Given Order." Discrete Dynamics in Nature and Society 2020 (July 18, 2020): 1–9. http://dx.doi.org/10.1155/2020/1217567.

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The variable connectivity index, introduced by the chemist Milan Randić in the first quarter of 1990s, for a graph G is defined as ∑vw∈EGdv+γdw+γ−1/2, where γ is a non-negative real number and dw is the degree of a vertex w in G. We call this index as the variable Randić index and denote it by Rvγ. In this paper, we show that the graph created from the star graph of order n by adding an edge has the minimum Rvγ value among all unicyclic graphs of a fixed order n, for every n≥4 and γ≥0.

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Ratnasari, Lucia, Sri Wahyuni, Yeni Susanti, and Diah Junia Eksi Palupi. "Total Edge Irregularity Strength of q Tuple Book Graphs." Journal of Mathematics Research 13, no. 1 (January 11, 2021): 16. http://dx.doi.org/10.5539/jmr.v13n1p16.

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Let G(V, E) be a simple, undirected, and finite graph with a vertex set V and an edge set E. An edge irregular total k-labelling is a function f from the set V \cup E to the set of non-negative integer set (1, 2, ... , k) such that any two different edges in E have distinct weights. The weight of edge xy is defined as the sum of the label of vertex x, the label of vertex y and the label of edge xy. The minimum k for which the graph G can be labelled by an edge irregular total k-labelling is called the total edge irregularity strength of G, denoted by tes(G). We have constructed the formula of an edge irregular total k-labelling and determined the total edge irregularity strength of triple book graphs, quadruplet book graphs and quintuplet book graphs. In this paper, we construct an edge irregular total of k-labelling and show the exact value of the total edge irregularity strength of q tuple book graphs.

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Ratnasari, Lucia, Sri Wahyuni, Yeni Susanti, and Diah Junia Eksi Palupi. "Total Edge Irregularity Strength of q Tuple Book Graphs." Journal of Mathematics Research 13, no. 1 (January 11, 2021): 16. http://dx.doi.org/10.5539/jmr.v13n1p16.

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Let G(V, E) be a simple, undirected, and finite graph with a vertex set V and an edge set E. An edge irregular total k-labelling is a function f from the set V \cup E to the set of non-negative integer set (1, 2, ... , k) such that any two different edges in E have distinct weights. The weight of edge xy is defined as the sum of the label of vertex x, the label of vertex y and the label of edge xy. The minimum k for which the graph G can be labelled by an edge irregular total k-labelling is called the total edge irregularity strength of G, denoted by tes(G). We have constructed the formula of an edge irregular total k-labelling and determined the total edge irregularity strength of triple book graphs, quadruplet book graphs and quintuplet book graphs. In this paper, we construct an edge irregular total of k-labelling and show the exact value of the total edge irregularity strength of q tuple book graphs.

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Naduvath, Sudev. "A study on the modular sumset labeling of graphs." Discrete Mathematics, Algorithms and Applications 09, no. 03 (May 2, 2017): 1750039. http://dx.doi.org/10.1142/s1793830917500392.

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For a positive integer [Formula: see text], let [Formula: see text] be the set of all non-negative integers modulo [Formula: see text] and [Formula: see text] be its power set. A modular sumset valuation or a modular sumset labeling of a given graph [Formula: see text] is an injective function [Formula: see text] such that the induced function [Formula: see text] defined by [Formula: see text]. A modular sumset indexer of a graph [Formula: see text] is an injective modular sumset valued function [Formula: see text] such that the induced function [Formula: see text] is also injective. In this paper, some properties and characteristics of this type of modular sumset labeling of graphs are being studied.

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30

Naikoo, T. A., U. Samee, S. Pirzada, and Bilal A. Rather. "On degree sets in k-partite graphs." Acta Universitatis Sapientiae, Informatica 12, no. 2 (December 1, 2020): 251–59. http://dx.doi.org/10.2478/ausi-2020-0015.

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Sudev, N. K., and K. A. Germina. "Some new results on strong integer additive set-indexers of graphs." Discrete Mathematics, Algorithms and Applications 07, no. 01 (February 2, 2015): 1450065. http://dx.doi.org/10.1142/s1793830914500657.

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Let ℕ0 be the set of all non-negative integers. An integer additive set-indexer (IASI) of a graph G is an injective function f : V(G) → 2ℕ0 such that the induced function gf : E(G) → 2ℕ0 defined by f+(uv) = f(u) + f(v) is also injective. An IASI is said to be k-uniform if |f+(e)| = k for all e ∈ E(G). In this paper, we introduce the notions of strong IASIs and initiate a study of the graphs which admit strong IASIs.

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32

Et. al., Meera Saraswathi,. "Radio Mean Labeling Of Paths And Its Total Graph." Turkish Journal of Computer and Mathematics Education (TURCOMAT) 12, no. 1S (April 11, 2021): 343–50. http://dx.doi.org/10.17762/turcomat.v12i1s.1843.

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A graph labeling problem is an assignment of labels to the vertices or edges (or both) of a graph G satisfying some mathematical condition. Radio Mean Labeling, a vertex-labeling of graphs with non-negative integers has a significant application in the study of problems related to radio channel assignment. The maximum label used in a radio mean labeling is called its span, and the lowest possible span of a radio mean labeling is called the radio mean number of a graph. In this paper, we obtain the radio mean number of paths and total graph of paths.

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Pirzada, Shariefuddin, Bilal Ahmad Chat, and Uma Tul Samee. "On multigraphic and potentially multigraphic sequences." Acta Universitatis Sapientiae, Informatica 9, no. 1 (July 26, 2017): 35–47. http://dx.doi.org/10.1515/ausi-2017-0003.

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AbstractAn r-graph(or a multigraph) is a loopless graph in which no two vertices are joined by more than r edges. An r-complete graph on n vertices, denoted by Kn(r), is an r-graph on n vertices in which each pair of vertices is joined by exactly r edges. A non-increasing sequence π = (d1,d2,..., dn) of non-negative integers is said to be r-graphic if it is realizable by an r-graph on n vertices. An r-graphic sequence π is said to be potentially SL;M(r)-graphic if it has a realization containing SL;M(r)as a subgraph. We obtain conditions for an r-graphic sequence to be potentially S(r) L;M-graphic. These are generalizations from split graphs to p-tuple r-split graph.

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McKAY, BRENDAN D. "Subgraphs of Dense Random Graphs with Specified Degrees." Combinatorics, Probability and Computing 20, no. 3 (January 27, 2011): 413–33. http://dx.doi.org/10.1017/s0963548311000034.

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Let d = (d1, d2, . . ., dn) be a vector of non-negative integers with even sum. We prove some basic facts about the structure of a random graph with degree sequence d, including the probability of a given subgraph or induced subgraph.Although there are many results of this kind, they are restricted to the sparse case with only a few exceptions. Our focus is instead on the case where the average degree is approximately a constant fraction of n.Our approach is the multidimensional saddle-point method. This extends the enumerative work of McKay and Wormald (1990) and is analogous to the theory developed for bipartite graphs by Greenhill and McKay (2009).

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WU, BANG YE. "FULLY POLYNOMIAL-TIME APPROXIMATION SCHEMES FOR THE MAX–MIN CONNECTED PARTITION PROBLEM ON INTERVAL GRAPHS." Discrete Mathematics, Algorithms and Applications 04, no. 01 (March 2012): 1250005. http://dx.doi.org/10.1142/s179383091250005x.

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We study how to partition an interval graph with non-negative vertex weights into k connected subgraphs such that the minimum total weight of any part of the partition is maximized. For k = 2, it is shown that for any ε > 0, a (1 + ε)-approximation can be found in O((1/ε)n3) time, i.e., it admits a fully polynomial-time approximation scheme (FPTAS). For any fixed k > 2, the problem also admits an FPTAS when restricted to k-connected interval graphs.

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Togan, Muge, Aysun Yurttas, Utkum Sanli, Feriha Celik, and Ismail Cangual. "Inverse problem for Bell index." Filomat 34, no. 2 (2020): 615–21. http://dx.doi.org/10.2298/fil2002615t.

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Due to their applications in many branches of science, topological graph indices are becoming more popular every day. Especially as one can model chemical molecules by graphs to obtain valuable information about the molecules using solely mathematical calculations on the graph. The inverse problem for topological graph indices is a recent problem proposed by Gutman and is about the existence of a graph having its index value equal to a given non-negative integer. In this paper, the inverse problem for Bell index which is one of the irregularity indices is solved. Also a recently defined graph invariant called omega invariant is used to obtain several properties related to the Bell index.

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37

Pirzada, S., U. Samee, and T. A. Naikoo. "Tournaments, oriented graphs and football sequences." Acta Universitatis Sapientiae, Mathematica 9, no. 1 (August 1, 2017): 213–23. http://dx.doi.org/10.1515/ausm-2017-0014.

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Abstract Consider the result of a soccer league competition where n teams play each other exactly once. A team gets three points for each win and one point for each draw. The total score obtained by each team vi is called the f-score of vi and is denoted by fi. The sequences of all f-scores $\left[ {{\rm{f}}_{\rm{i}} } \right]_{{\rm{i}} = 1}^{\rm{n}} $ arranged in non-decreasing order is called the f-score sequence of the competition. We raise the following problem: Which sequences of non-negative integers in non-decreasing order is a football sequence, that is the outcome of a soccer league competition. We model such a competition by an oriented graph with teams represented by vertices in which the teams play each other once, with an arc from team u to team v if and only if u defeats v. We obtain some necessary conditions for football sequences and some characterizations under restrictions.

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38

Simson, Daniel, and Katarzyna Zając. "Inflation algorithm for loop-free non-negative edge-bipartite graphs of corank at least two." Linear Algebra and its Applications 524 (July 2017): 109–52. http://dx.doi.org/10.1016/j.laa.2017.02.021.

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39

Du, Jinzhi, and Jianhua Yin. "A further result on the potential-Ramsey number of G1 and G2." Filomat 33, no. 6 (2019): 1605–17. http://dx.doi.org/10.2298/fil1906605d.

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A non-increasing sequence ? = (d1,. . ., dn) of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices. In this case, G is referred to as a realization of ?. Given a graph H, a graphic sequence ? is potentially H-graphic if ? has a realization containing H as a subgraph. Busch et al. (Graphs Combin., 30(2014)847-859) considered a degree sequence analogue to classical graph Ramsey number as follows: for graphs G1 and G2, the potential-Ramsey number rpot(G1,G2) is the smallest non-negative integer k such that for any k-term graphic sequence ?, either ? is potentially G1-graphic or the complementary sequence ? = (k - 1 - dk,..., k - 1 - d1) is potentially G2-graphic. They also gave a lower bound on rpot(G;Kr+1) for a number of choices of G and determined the exact values for rpot(Kn;Kr+1), rpot(Cn;Kr+1) and rpot(Pn,Kr+1). In this paper, we will extend the complete graph Kr+1 to the complete split graph Sr,s = Kr ? Ks. Clearly, Sr,1 = Kr+1. We first give a lower bound on rpot(G, Sr,s) for a number of choices of G, and then determine the exact values for rpot(Cn, Sr,s) and rpot(Pn, Sr,s).

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Abudayah, Mohammad, Omar Alomari, and Torsten Sander. "On the N-spectrum of oriented graphs." Open Mathematics 18, no. 1 (June 4, 2020): 486–95. http://dx.doi.org/10.1515/math-2020-0167.

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Abstract Given any digraph D, its non-negative spectrum (or N-spectrum, shortly) consists of the eigenvalues of the matrix AA T , where A is the adjacency matrix of D. In this study, we relate the classical spectrum of undirected graphs to the N-spectrum of their oriented counterparts, permitting us to derive spectral bounds. Moreover, we study the spectral effects caused by certain modifications of a given digraph.

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Mohamad, Siti Nurul Fitriah, Roslan Hasni, Florentin Smarandache, and Binyamin Yusoff. "Novel Concept of Energy in Bipolar Single-Valued Neutrosophic Graphs with Applications." Axioms 10, no. 3 (July 29, 2021): 172. http://dx.doi.org/10.3390/axioms10030172.

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The energy of a graph is defined as the sum of the absolute values of its eigenvalues. Recently, there has been a lot of interest in graph energy research. Previous literature has suggested integrating energy, Laplacian energy, and signless Laplacian energy with single-valued neutrosophic graphs (SVNGs). This integration is used to solve problems that are characterized by indeterminate and inconsistent information. However, when the information is endowed with both positive and negative uncertainty, then bipolar single-valued neutrosophic sets (BSVNs) constitute an appropriate knowledge representation of this framework. A BSVNs is a generalized bipolar fuzzy structure that deals with positive and negative uncertainty in real-life problems with a larger domain. In contrast to the previous study, which directly used truth and indeterminate and false membership, this paper proposes integrating energy, Laplacian energy, and signless Laplacian energy with BSVNs to graph structure considering the positive and negative membership degree to greatly improve decisions in certain problems. Moreover, this paper intends to elaborate on characteristics of eigenvalues, upper and lower bound of energy, Laplacian energy, and signless Laplacian energy. We introduced the concept of a bipolar single-valued neutrosophic graph (BSVNG) for an energy graph and discussed its relevant ideas with the help of examples. Furthermore, the significance of using bipolar concepts over non-bipolar concepts is compared numerically. Finally, the application of energy, Laplacian energy, and signless Laplacian energy in BSVNG are demonstrated in selecting renewable energy sources, while optimal selection is suggested to illustrate the proposed method. This indicates the usefulness and practicality of this proposed approach in real life.

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GUAN, YU, and GUANGHUI XU. "ALGEBRAIC CONNECTIVITY OF WEIGHED GRAPHS UNDER SHIFTING COMPONENTS." Discrete Mathematics, Algorithms and Applications 02, no. 03 (September 2010): 263–75. http://dx.doi.org/10.1142/s1793830910000656.

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Integrating quadratic form of algebraic connectivity and Perron value of bottleneck matrices, we investigate how the algebraic connectivity of a connected weighted graph behaves under shifting components. Generally speaking, when shift components not containing characteristic vertex from less positive (larger negative) valuation vertices to larger positive (less negative) valuation vertices, or reduce weights of some edges, or add some new blocks, its algebraic connectivity is nonincreasing; when shift components along paths from blocks to other block closer to characteristic block (vertex), or increase weights of some edges, or delete some blocks, its algebraic connectivity is non-decreasing. Therefore, algebraic connectivity could be regarded as a measure of central tendency about blocks of a connected weighted graph with characteristic block (vertex) as its center.

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TRALDI, LORENZO. "A BRACKET POLYNOMIAL FOR GRAPHS, II: LINKS, EULER CIRCUITS AND MARKED GRAPHS." Journal of Knot Theory and Its Ramifications 19, no. 04 (April 2010): 547–86. http://dx.doi.org/10.1142/s0218216510007978.

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Let D be an oriented classical or virtual link diagram with directed universe [Formula: see text]. Let C denote a set of directed Euler circuits, one in each connected component of U. There is then an associated looped interlacement graph [Formula: see text] whose construction involves very little geometric information about the way D is drawn in the plane; consequently [Formula: see text] is different from other combinatorial structures associated with classical link diagrams, like the checkerboard graph, which can be difficult to extend to arbitrary virtual links. [Formula: see text] is determined by three things: the structure of [Formula: see text] as a 2-in, 2-out digraph, the distinction between crossings that make a positive contribution to the writhe and those that make a negative contribution, and the relationship between C and the directed circuits in [Formula: see text] arising from the link components; this relationship is indicated by marking the vertices where C does not follow the incident link component(s). We introduce a bracket polynomial for arbitrary marked graphs, defined using either a formula involving matrix nullities or a recursion involving the local complement and pivot operations; the marked-graph bracket of [Formula: see text] is the same as the Kauffman bracket of D. This provides a unified combinatorial description of the Jones polynomial that applies seamlessly to both classical and non-classical virtual links.

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Asgharsharghi, L., S. M. Sheikholeslami, and L. Volkmann. "A note on the 2-rainbow bondage numbers in graphs." Asian-European Journal of Mathematics 09, no. 01 (February 22, 2016): 1650013. http://dx.doi.org/10.1142/s1793557116500133.

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A 2-rainbow dominating function (2RDF) of a graph [Formula: see text] is a function [Formula: see text] from the vertex set [Formula: see text] to the set of all subsets of the set [Formula: see text] such that for any vertex [Formula: see text] with [Formula: see text], the condition [Formula: see text] is fulfilled. The weight of a 2RDF [Formula: see text] is the value [Formula: see text]. The [Formula: see text]-rainbow domination number of a graph [Formula: see text], denoted by [Formula: see text], is the minimum weight of a 2RDF of [Formula: see text]. The rainbow bondage number [Formula: see text] of a graph [Formula: see text] with maximum degree at least two is the minimum cardinality of all sets [Formula: see text] for which [Formula: see text]. Dehgardi, Sheikholeslami and Volkmann, [The [Formula: see text]-rainbow bondage number of a graph, Discrete Appl. Math. 174 (2014) 133–139] proved that the rainbow bondage number of a planar graph does not exceed 15. In this paper, we generalize their result for graphs which admit a [Formula: see text]-cell embedding on a surface with non-negative Euler characteristic.

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45

Sudev, N. K., K. A. Germina, and K. P. Chithra. "Weak integer additive set-labeled graphs: A creative review." Asian-European Journal of Mathematics 08, no. 03 (September 2015): 1550052. http://dx.doi.org/10.1142/s1793557115500527.

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For a non-empty ground set [Formula: see text], finite or infinite, the set-valuation or set-labeling of a given graph [Formula: see text] is an injective function [Formula: see text], where [Formula: see text] is the power set of the set [Formula: see text]. A set-valuation or a set-labeling of a graph [Formula: see text] is an injective set-valued function [Formula: see text] such that the induced function [Formula: see text] is defined by [Formula: see text] for every [Formula: see text], where [Formula: see text] is a binary operation on sets. Let [Formula: see text] be the set of all non-negative integers and [Formula: see text] be its power set. An integer additive set-labeling (IASL) is defined as an injective function [Formula: see text] such that the induced function [Formula: see text] is defined by [Formula: see text]. An IASL [Formula: see text] is said to be an integer additive set-indexer if [Formula: see text] is also injective. A weak IASL is an IASL [Formula: see text] such that [Formula: see text]. In this paper, critical and creative review of certain studies made on the concepts and properties of weak integer additive set-valued graphs is intended.

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FARSI, CARLA, ELIZABETH GILLASPY, PALLE JORGENSEN, SOORAN KANG, and JUDITH PACKER. "Monic representations of finite higher-rank graphs." Ergodic Theory and Dynamical Systems 40, no. 5 (September 6, 2018): 1238–67. http://dx.doi.org/10.1017/etds.2018.79.

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In this paper, we define the notion of monic representation for the$C^{\ast }$-algebras of finite higher-rank graphs with no sources, and we undertake a comprehensive study of them. Monic representations are the representations that, when restricted to the commutative$C^{\ast }$-algebra of the continuous functions on the infinite path space, admit a cyclic vector. We link monic representations to the$\unicode[STIX]{x1D6EC}$-semibranching representations previously studied by Farsi, Gillaspy, Kang and Packer (Separable representations, KMS states, and wavelets for higher-rank graphs.J. Math. Anal. Appl. 434 (2015), 241–270) and also provide a universal representation model for non-negative monic representations.

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Minello, Giorgia, Luca Rossi, and Andrea Torsello. "On the von Neumann entropy of graphs." Journal of Complex Networks 7, no. 4 (November 23, 2018): 491–514. http://dx.doi.org/10.1093/comnet/cny028.

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Abstract The von Neumann entropy of a graph is a spectral complexity measure that has recently found applications in complex networks analysis and pattern recognition. Two variants of the von Neumann entropy exist based on the graph Laplacian and normalized graph Laplacian, respectively. Due to its computational complexity, previous works have proposed to approximate the von Neumann entropy, effectively reducing it to the computation of simple node degree statistics. Unfortunately, a number of issues surrounding the von Neumann entropy remain unsolved to date, including the interpretation of this spectral measure in terms of structural patterns, understanding the relation between its two variants and evaluating the quality of the corresponding approximations. In this article, we aim to answer these questions by first analysing and comparing the quadratic approximations of the two variants and then performing an extensive set of experiments on both synthetic and real-world graphs. We find that (1) the two entropies lead to the emergence of similar structures, but with some significant differences; (2) the correlation between them ranges from weakly positive to strongly negative, depending on the topology of the underlying graph; (3) the quadratic approximations fail to capture the presence of non-trivial structural patterns that seem to influence the value of the exact entropies; and (4) the quality of the approximations, as well as which variant of the von Neumann entropy is better approximated, depends on the topology of the underlying graph.

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BALOGH, JÓZSEF, JANE BUTTERFIELD, PING HU, JOHN LENZ, and DHRUV MUBAYI. "On the Chromatic Thresholds of Hypergraphs." Combinatorics, Probability and Computing 25, no. 2 (March 6, 2015): 172–212. http://dx.doi.org/10.1017/s0963548315000061.

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Let $\mathcal{F}$ be a family of r-uniform hypergraphs. The chromatic threshold of $\mathcal{F}$ is the infimum of all non-negative reals c such that the subfamily of $\mathcal{F}$ comprising hypergraphs H with minimum degree at least $c \binom{| V(H) |}{r-1}$ has bounded chromatic number. This parameter has a long history for graphs (r = 2), and in this paper we begin its systematic study for hypergraphs.Łuczak and Thomassé recently proved that the chromatic threshold of the so-called near bipartite graphs is zero, and our main contribution is to generalize this result to r-uniform hypergraphs. For this class of hypergraphs, we also show that the exact Turán number is achieved uniquely by the complete (r + 1)-partite hypergraph with nearly equal part sizes. This is one of very few infinite families of non-degenerate hypergraphs whose Turán number is determined exactly. In an attempt to generalize Thomassen's result that the chromatic threshold of triangle-free graphs is 1/3, we prove bounds for the chromatic threshold of the family of 3-uniform hypergraphs not containing {abc, abd, cde}, the so-called generalized triangle.In order to prove upper bounds we introduce the concept of fibre bundles, which can be thought of as a hypergraph analogue of directed graphs. This leads to the notion of fibre bundle dimension, a structural property of fibre bundles that is based on the idea of Vapnik–Chervonenkis dimension in hypergraphs. Our lower bounds follow from explicit constructions, many of which use a hypergraph analogue of the Kneser graph. Using methods from extremal set theory, we prove that these Kneser hypergraphs have unbounded chromatic number. This generalizes a result of Szemerédi for graphs and might be of independent interest. Many open problems remain.

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Mehra, Seema, and Puneet. "A characterization for topologically integer additive set-indexer of graphs." Discrete Mathematics, Algorithms and Applications 08, no. 01 (February 26, 2016): 1650012. http://dx.doi.org/10.1142/s1793830916500129.

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The aim of this paper is to introduce a new function that satisfies the property of topological integer additive set-indexer (Top-IASI) with minimum cardinality for pan, tadpole, path and shovel graphs. Let [Formula: see text] be a graph and [Formula: see text] is a nonempty set. If an injective function [Formula: see text] induced a new injective function [Formula: see text] defined by [Formula: see text] for every [Formula: see text] then [Formula: see text] is called set-indexer. If an injective function [Formula: see text] produced another injective function [Formula: see text] defined by [Formula: see text] for every [Formula: see text], where [Formula: see text] is the set of all non-negative integers and [Formula: see text] is its power set then [Formula: see text] is called an integer additive set-indexer (IASI). An IASI is called a Top-IASI if [Formula: see text] forms a topology.

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50

Gavars, Raivis, Einārs Netlis-Galejs, Jānis Artūrs Lazdiņš, and Ilmārs Kangro. "FLOYD–WARSHALL ALGORITHM FOR SHORTEST ROUTE CALCULATION BETWEEN LATVIA'S CITIES." HUMAN. ENVIRONMENT. TECHNOLOGIES. Proceedings of the Students International Scientific and Practical Conference, no. 23 (April 24, 2019): 31. http://dx.doi.org/10.17770/het2019.23.4397.

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The Floyd–Warshall algorithm is a good choice for computing paths between all pairs of vertices indense graphs, in which most or all pairs of vertices are connected by edges. For sparse graphs with non-negative edgeweights, a better choice is to use Dijkstra's algorithm from each possible starting vertex. Also, a very good thing is that thesolution is very accurate, when using a computer. In this paper, the authors tried to apply a solution using C++programming language to make possible many entries.

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