On irreducibility of eccentricity matrix of graphs and construction of $\epsilon-$equienergetic graphs

TL;DR

This paper investigates the spectral properties of eccentricity matrices of graphs, explores conditions for irreducibility, and constructs new families of epsilon-equienergetic and epsilon-integral graphs, advancing spectral graph theory.

Contribution

It introduces new results on the eccentricity spectrum of certain graph operations and constructs infinite families of epsilon-cospectral and epsilon-equienergetic graphs.

Findings

Identifies conditions for irreducibility of eccentricity matrices.

Constructs infinite families of epsilon-cospectral and epsilon-equienergetic graphs.

Provides new examples of epsilon-integral graphs.

Abstract

The eccentricity matrix $\epsilon(G)$, of a connected graph $G$ is obtained by retaining the maximum distance from each row and column of the distance matrix of $G$ and the other entries are assigned with 0. In this paper, we discuss the eccentricity spectrum of subdivision vertex (edge) join of regular graphs. Also, we obtain new families of graphs having irreducible or reducible eccentricity matrix. Furthermore, we use these results to construct infinitely many $\epsilon-$cospectral graph pairs as well as infinitely many pairs and triplets of $\epsilon-$cospectral $\epsilon-$equienergetic graphs. Moreover, we present some new family of $\epsilon-$integral graphs.