Eccentricity spectrum of join of central graphs and Eccentricity Wiener index of graphs

TL;DR

This paper studies the eccentricity spectrum and related indices of central graphs and their operations, providing bounds and constructing new classes of graphs with specific spectral properties.

Contribution

It introduces new spectral analysis of central graphs, explores their join operations, and establishes bounds for eccentricity-based indices, enriching graph spectral theory.

Findings

Derived the eccentricity spectrum and energy for central graphs.

Constructed new families of epsilon-cospectral and epsilon-equienergetic graphs.

Established bounds for eccentricity Wiener index and eccentricity energy.

Abstract

The eccentricity matrix of a simple connected graph is derived from its distance matrix by preserving the largest non-zero distance in each row and column, while the other entries are set to zero. This article examines the $\epsilon$-spectrum, $\epsilon$-energy, $\epsilon$-inertia and irreducibility of the central graph (respectively complement of the central graph) of a triangle-free regular graph(respectively regular graph). Also look into the $\epsilon-$spectrum and the irreducibility of different central graph operations, such as central vertex join, central edge join, and central vertex-edge join. We also examine the $\epsilon-$ energy of some specific graphs. These findings allow us to construct new families of $\epsilon$-cospectral graphs and non $\epsilon$-cospectral $\epsilon-$equienergetic graphs. Additionally, we investigate certain upper and lower bounds for the eccentricity Wiener index of graphs. Also, provide an upper bound for the eccentricity energy of a self-centered graph.