Spectral characterization of mixed extensions of small graphs

TL;DR

This paper characterizes graphs with at most three eigenvalues different from 0 and -1, focusing on mixed extensions of small graphs and identifying their spectral properties.

Contribution

It determines the class of graphs with limited eigenvalue diversity, extending the understanding of spectral graph theory for small and specific graph structures.

Findings

Graphs with at most three eigenvalues not equal to 0 or -1 are characterized.

Mixed extensions of small graphs and certain paths form the class of such graphs.

The spectral properties of these graphs are explicitly described.

Abstract

A mixed extension of a graph $G$ is a graph $H$ obtained from $G$ by replacing each vertex of $G$ by a clique or a coclique, where vertices of $H$ coming from different vertices of $G$ are adjacent if and only if the original vertices are adjacent in $G$. If $G$ has no more than three vertices, $H$ has all but at most three adjacency eigenvalues equal to $0$ or $-1$. In this paper we consider the converse problem, and determine the class $\cal G$ of all graphs with at most three eigenvalues unequal to $0$ and $-1$. Ignoring isolated vertices, we find that $\cal G$ consists of all mixed extensions of graphs on at most three vertices together with some particular mixed extensions of the paths $P_4$ and $P_5$.