Locating Eigenvalues of a Symmetric Matrix whose Graph is Unicyclic

R O Braga,R O Silva,V M Rodrigues

doi:10.5540/tcam.2021.022.04.00659

Abstract

We present a linear-time algorithm that computes in a given real interval the number of eigenvalues of any symmetric matrix whose underlying graph is unicyclic. The algorithm can be applied to vertex- and/or edge-weighted or unweighted unicyclic graphs. We apply the algorithm to obtain some general results on the spectrum of a generalized sun graph for certain matrix representations which include the Laplacian, normalized Laplacian and signless Laplacian matrices.

Highlights

Given a real symmetric matrix M we consider the underlying graph of M, i.e., the simple graph G(M) whose vertices correspond to rows of M with two vertices vi and v j adjacent whenever the (i, j)th entry of M is nonzero
For the purpose of this work each diagonal entry mii of M is assigned as a weight to the vertex vi of G(M) and each off-diagonal entry mi j is assigned as an edge weight to the edge ei j connecting vertices vi and v j of G(M)
Besides preserving the practicality of the previous algorithms for locating eigenvalues of unicyclic graphs represented by their adjacency matrices, our method has the advantage of allowing vertex- and/or edge-weighted unicyclic graphs represented by any symmetric matrix whose underlying graph is unicyclic

Summary

INTRODUCTION

Given a real symmetric matrix M we consider the underlying graph of M, i.e., the simple graph G(M) whose vertices correspond to rows (or columns) of M with two vertices vi and v j adjacent whenever the (i, j)th entry of M is nonzero. In 2017 Braga, Rodrigues and Trevisan [4] presented an algorithm that computes the number of eigenvalues in a given real interval of an unweighted unicyclic graph represented by its adjacency matrix, called DiagonalizeUnicyclic. Their algorithm was the main tool applied in [2] to reveal infinite families of integral unicyclic graphs, i.e. unicyclic graphs where all the eigenvalues of their adjacency matrices are integers. Besides preserving the practicality of the previous algorithms for locating eigenvalues of unicyclic graphs represented by their adjacency matrices, our method has the advantage of allowing vertex- and/or edge-weighted unicyclic graphs represented by any symmetric matrix whose underlying graph is unicyclic.

ALGORITHM DIAGSMUNICYCLIC

GENERALIZED SUN GRAPHS

CONCLUSION