Hyperbolicity on Graph Operators

A graph operator is a mapping F : Gamma → Gamma 0 , where Gamma and Gamma 0 are families of graphs. The different kinds of graph operators are an important topic in Discrete Mathematics and its applications. The symmetry of this operations allows us to prove inequalities relating the hyperbolicity constants of a graph G and its graph operators: line graph, Lambda(G); subdivision graph, S(G); total graph, T(G); and the operators R(G) and Q(G). In particular, we get relationships such as delta(G) ≤ delta(R(G)) ≤ delta(G) + 1/2, delta(Lambda(G)) ≤ delta(Q(G)) ≤ delta(Lambda(G)) + 1/2, delta(S(G)) ≤ 2delta(R(G)) ≤ delta(S(G)) + 1 and delta(R(G)) − 1/2 ≤ delta(Lambda(G)) ≤ 5delta(R(G)) + 5/2 for every graph which is not a tree. Moreover, we also derive some inequalities for the Gromov product and the Gromov product restricted to vertices.Supported in part by two grants from Ministerio de Economía y Competitividad, Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) (MTM2016-78227-C2-1-P and MTM2015-69323-REDT), Spain

Harmonic index and harmonic polynomial on graph operations

Some years ago, the harmonic polynomial was introduced to study the harmonic topological index. Here, using this polynomial, we obtain several properties of the harmonic index of many classical symmetric operations of graphs: Cartesian product, corona product, join, Cartesian sum and lexicographic product. Some upper and lower bounds for the harmonic indices of these operations of graphs, in terms of related indices, are derived from known bounds on the integral of a product on nonnegative convex functions. Besides, we provide an algorithm that computes the harmonic polynomial with complexity O(n 2 ).This work was supported in part by two grants from Ministerio de Economía y Competititvidad, Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) (MTM2016-78227-C2-1-P and MTM2017-90584-REDT), Spain

Mathematical Properties on the Hyperbolicity of Interval Graphs

Gromov hyperbolicity is an interesting geometric property, and so it is natural to study it in the context of geometric graphs. In particular, we are interested in interval and indifference graphs, which are important classes of intersection and Euclidean graphs, respectively. Interval graphs (with a very weak hypothesis) and indifference graphs are hyperbolic. In this paper, we give a sharp bound for their hyperbolicity constants. The main result in this paper is the study of the hyperbolicity constant of every interval graph with edges of length 1. Moreover, we obtain sharp estimates for the hyperbolicity constant of the complement of any interval graph with edges of length 1.This paper was supported in part by a grant from CONACYT (FOMIX-CONACyT-UAGro 249818), México and by two grants from the Ministerio de Economía y Competitividad, Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) (MTM2016-78227-C2-1-P and MTM2015-69323-REDT), Spain. We would like to thank the referees for their careful reading of the manuscript and several useful comments that have helped us to improve the presentation of the paper