Locating eigenvalues of unicyclic graphs

Rodrigo Braga,Virgínia Rodrigues,Vilmar Trevisan

doi:10.2298/aadm1702273b

Rodrigo Braga, Virgínia Rodrigues + Show 1 more

Open Access

https://doi.org/10.2298/aadm1702273b

Copy DOI

Abstract

We present a linear time algorithm that computes the number of eigenvalues of a unicyclic graph in a given real interval. It operates directly on the graph, so that the matrix is not needed explicitly. The algorithm is applied to study the multiplicities of eigenvalues of closed caterpillars, obtain the spectrum of balanced closed caterpillars and give sufficient conditions for these graphs to be non-integral. We also use our method to study the distribution of eigenvalues of unicyclic graphs formed by adding a fixed number of copies of a path to each node in a cycle. We show that they are not integral graphs.

Highlights

Let G be a simple undirected graph with vertices v1, ..., vn
In [6], Fritscher et al used the algorithm adapted for the Laplacian matrix of a tree, defined as L = D − A, where D is the diagonal matrix whose (i, i)−entry is the degree of vertex vi and A is the adjacency matrix of the tree. They applied the algorithm to derive a new upper bound on the sum of the k largest Laplacian eigenvalues of a tree
A natural extension of the algorithm for locating eigenvalues of trees is to consider unicyclic graphs. This is the aim of this work, where we present, in Section 2, an algorithm to determine the number of eigenvalues of a unicyclic graph in a given real interval

Summary

INTRODUCTION

Let G be a simple undirected graph with vertices v1, ..., vn. The adjacency matrix A = (aij) of G is the 0 − 1 matrix of order n, where aij = 1 if and only if vi is adjacent to vj. In [9], it was applied to study the multiplicities of the eigenvalues of caterpillars, which they considered as trees formed by taking a path Pb on b 2 vertices and adding at least one pendant vertex to each node in the path. In [6], Fritscher et al used the algorithm adapted for the Laplacian matrix of a tree, defined as L = D − A, where D is the diagonal matrix whose (i, i)−entry is the degree of vertex vi and A is the adjacency matrix of the tree They applied the algorithm to derive a new upper bound on the sum of the k largest Laplacian eigenvalues of a tree.

ALGORITHM FOR LOCATING EIGENVALUES OF UNICYCLIC GRAPHS

EXAMPLE

APPLICATIONS

CONCLUDING REMARKS