63 research outputs found

    Chromaticity Of Certain K4-Homeomorphs

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    The chromaticity of graphs is the term used referring to the question of chromatic equivalence and chromatic uniqueness of graphs. Since the arousal of the interest on the chromatically equivalent and chromatically unique graphs, various concepts and results under the said areas of research have been discovered and many families of such graphs have been obtained. The purpose of this thesis is to contribute new results on the chromatic equivalence and chromatic uniqueness of graphs, specifically, K4-homeomorphs

    Chromatic uniqueness of a family of K4-homeomorphs

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    AbstractWe discuss the chromaticity of one family of K4-homeomorphs which has girth 7, and give sufficient and necessary condition for the graphs in the family to be chromatically unique

    Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions

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    I show that there exist universal constants C(r)<C(r) < \infty such that, for all loopless graphs GG of maximum degree r\le r, the zeros (real or complex) of the chromatic polynomial PG(q)P_G(q) lie in the disc q<C(r)|q| < C(r). Furthermore, C(r)7.963906...rC(r) \le 7.963906... r. This result is a corollary of a more general result on the zeros of the Potts-model partition function ZG(q,ve)Z_G(q, {v_e}) in the complex antiferromagnetic regime 1+ve1|1 + v_e| \le 1. The proof is based on a transformation of the Whitney-Tutte-Fortuin-Kasteleyn representation of ZG(q,ve)Z_G(q, {v_e}) to a polymer gas, followed by verification of the Dobrushin-Koteck\'y-Preiss condition for nonvanishing of a polymer-model partition function. I also show that, for all loopless graphs GG of second-largest degree r\le r, the zeros of PG(q)P_G(q) lie in the disc q<C(r)+1|q| < C(r) + 1. Along the way, I give a simple proof of a generalized (multivariate) Brown-Colbourn conjecture on the zeros of the reliability polynomial for the special case of series-parallel graphs.Comment: 47 pages (LaTeX). Revised version contains slightly simplified proofs of Propositions 4.2 and 4.5. Version 3 fixes a silly error in my proof of Proposition 4.1, and adds related discussion. To appear in Combinatorics, Probability & Computin

    Classifying Derived Voltage Graphs

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    Gross and Tucker’s voltage graph construction assigns group elements as weights to the edges of an oriented graph. This construction provides a blueprint for inducing graph covers. Thomas Zaslavsky studies the criteria for balance in voltage graphs. This project primarily examines the relationship between the group structure of the set of all possible assignments of a group to a graph, including the balanced subgroup, and the isomorphism classes of covering graphs. We examine connectedness, planarity, and chromatic number in the derived graph. Lastly we explain the future research possibilities involving the fundamental group

    Symmetric Product Graphs

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    We describe four types of hyperspace graphs; namely, the simultaneous and nonsimultaneous symmetric product graphs, as well as their respective layers. These hyperspace graphs are meant to be analogous to the concepts of hyperspaces in topology, in that they are constructed by taking in another graph as an input in the construction of the hyperspace graph. We establish subgraph relationship between these graphs and establish some properties on the orders and sizes of the graphs, as well as on the degrees of the individual vertices of these graphs. We establish that these graphs are connected (providing that the input graph is connected), and provide a categorization of the graphs G for which the second symmetric product graphs are planar. We investigate the chromatic numbers and hamiltonicity of some of these graph products. We also provide a categorization for the distances between any pair of vertices in the symmetric product graphs. We conclude by discussing a couple of different unanswered questions that could be addressed in the future

    Advances in Discrete Applied Mathematics and Graph Theory

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    The present reprint contains twelve papers published in the Special Issue “Advances in Discrete Applied Mathematics and Graph Theory, 2021” of the MDPI Mathematics journal, which cover a wide range of topics connected to the theory and applications of Graph Theory and Discrete Applied Mathematics. The focus of the majority of papers is on recent advances in graph theory and applications in chemical graph theory. In particular, the topics studied include bipartite and multipartite Ramsey numbers, graph coloring and chromatic numbers, several varieties of domination (Double Roman, Quasi-Total Roman, Total 3-Roman) and two graph indices of interest in chemical graph theory (Sombor index, generalized ABC index), as well as hyperspaces of graphs and local inclusive distance vertex irregular graphs
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