Iterated line graphs with only negative eigenvalues $-2$, their complements and energy

TL;DR

This paper characterizes iterated line graphs with all eigenvalues equal to -2, explores their spectral and energy properties, and introduces methods to analyze spectra of complex graph compositions.

Contribution

It provides a complete characterization of certain iterated line graphs with uniform negative eigenvalues and analyzes their energy and spectral properties, including complements and generalized compositions.

Findings

Iterated line graphs with all eigenvalues -2 are characterized.

Spectra and energy of these graphs and their complements are determined.

A new method using quotient matrices for spectra of generalized graph compositions is introduced.

Abstract

The graphs with all equal negative or positive eigenvalues are special kind in the spectral graph theory. In this article, several iterated line graphs $\mathcal{L}^k(G)$ with all equal negative eigenvalues $-2$ are characterized for $k\ge 1$ and their energy consequences are presented. Also, the spectra and the energy of complement of these graphs are obtained, interestingly they have exactly two positive eigenvalues with different multiplicities. Moreover, we characterize a large class of equienergetic graphs which generalize some of the existing results. There are two different quotient matrices defined for an equitable partition of $H$-join (generalized composition) of regular graphs to find the spectrum (partial) of adjacency matrix, Laplacian matrix and signless Laplacian matrix, it has been proved that these two quotient matrices give the same respective spectrum of graphs.