Spectral properties of $\mathcal{C}$-graphs

TL;DR

This paper explores the spectral properties of a specific class of cographs called $$-graphs, deriving their inertia, eigenvalue-free sets, and characteristic polynomial formulas based on their creation sequences.

Contribution

It introduces a method to relate creation sequences of $$-graphs to their spectral properties, including inertia and characteristic polynomial, providing new analytical tools for this graph class.

Findings

Inertia of $$-graphs can be explicitly determined from their creation sequences.

An extended eigenvalue-free set for $$-graphs is identified.

An exact formula for the characteristic polynomial of $$-graphs is derived.

Abstract

Assumed to be undirected, simple, and connected are all of the graphs in this study, and adjacency matrix $A$ serves as the associated matrix. In this paper we show that it is possible to relate a creation sequence for a type of cographs (we call it $\mathcal{C}$-graphs). Those cographs can be defined by a finite sequence of natural numbers. Using that sequence we obtain the inertia of the cograph under consideration. An extended eigenvalue-free set from $(-1,0)$ to $\big{[}\frac{-1-\sqrt{2}}{2}, -1)\cup (-1, 0) \cup (0, \frac{-1+\sqrt{2}}{2}\alpha_{min}\big{]}$, (where $\alpha_{min}\geq1$ is the smallest integer of the creation sequence) is obtained for the cographs under consideration. Additionally, an exact formula is found for the characteristic polynomial.