Partial Characterization of Graphs Having a Single Large Laplacian Eigenvalue

The parameter σ(G) of a graph G stands for the number of Laplacian eigenvalues greater than or equal to the average degree of G. In this work, we address the problem of characterizing those graphs G having σ(G) = 1. Our conjecture is that these graphs are stars plus a (possible empty) set of isolated vertices. We establish a link between σ(G) and the number of anticomponents of G. As a by-product, we present some results which support the conjecture, by restricting our analysis to cographs, forests, and split graphs.Fil: Allem, L. Emilio. Universidade Federal do Rio Grande do Sul; BrasilFil: Cafure, Antonio Artemio. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Dratman, Ezequiel. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Grippo, Luciano Norberto. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Safe, Martin Dario. Universidad Nacional del Sur; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Trevisan, Vilmar. Universidade Federal do Rio Grande do Sul; Brasi

Computing the Determinant of the Distance Matrix of a Bicyclic Graph

Let G be a connected graph with vertex set V = {v1, ..., vn}. The distance d(vi, vj) between two vertices vi and vj is the number of edges of a shortest path linking them. The distance matrix of G is the n × n matrix such that its (i, j)-entry is equal to d(vi, vj). A formula to compute the determinant of this matrix in terms of the number of vertices was found when the graph either is a tree or is a unicyclic graph. For a byciclic graph, the determinant is known in the case where the cycles have no common edges. In this paper, we present some advances for the remaining cases; i.e., when the cycles share at least one edge. We also present a conjecture for the unsolved cases.Fil: Dratman, Ezequiel. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaFil: Grippo, Luciano Norberto. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaFil: Safe, Martin Dario. Universidad Nacional del Sur. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: da Silva Jr., Celso M. Centro Federal de Educação Tecnológica; BrasilFil: Del Vecchio, Renata R.. Universidade Federal Fluminense; BrasilLAGOS 2019: X Latin and American Algorithms, Graphs and Optimization SymposiumBelo HorizonteBrasilUniversidad Federal de Minas Gerai