Enide Andrade

On the dominating induced matching problem: Spectral results and sharp bounds

Abstract A matching M is a dominating induced matching of a graph if every edge is either in M or has a common end-vertex with exactly one edge in M . The extremal graphs on the number of edges with dominating induced matchings are characterized by its Laplacian spectrum and its principal Laplacian eigenvector. Adjacency, Laplacian and signless Laplacian spectral bounds on the cardinality of dominating induced matchings are obtained for arbitrary graphs. Moreover, it is shown that some of these bounds are sharp and examples of graphs attaining the corresponding bounds are given.

Bethe graphs attached to the vertices of a connected graph: a spectral approach

A weighted Bethe graph B is obtained from a weighted generalized Bethe tree by identifying each set of children with the vertices of a graph belonging to a family F of graphs. The operation of identifying the root vertex of each of r weighted Bethe graphs to the vertices of a connected graph of order r is introduced as the -concatenation of a family of r weighted Bethe graphs. It is shown that the Laplacian eigenvalues (when F has arbitrary graphs) as well as the signless Laplacian and adjacency eigenvalues (when the graphs in F are all regular) of the -concatenation of a family of weighted Bethe graphs can be computed (in a unified way) using the stable and low computational cost methods available for the determination of the eigenvalues of symmetric tridiagonal matrices. Unlike the previous results already obtained on this topic, the more general context of families of distinct weighted Bethe graphs is herein considered.

Combinatorial Perron values of trees and bottleneck matrices

The algebraic connectivity a(G) of a graph G is an important parameter, defined as the second-smallest eigenvalue of the Laplacian matrix of G. If T is a tree, a(T) is closely related to the Perron values (spectral radius) of so-called bottleneck matrices of subtrees of T. In this setting, we introduce a new parameter called the combinatorial Perron value . This value is a lower bound on the Perron value of such subtrees; typically is a good approximation to . We compute exact values of for certain special subtrees. Moreover, some results concerning when the tree is modified are established, and it is shown that, among trees with given distance vector (from the root), is maximized for caterpillars.