Graphs that have a weighted adjacency matrix with spectrum $\{\lambda_1^{n-2}, \lambda_1^2\}$

TL;DR

This paper characterizes specific weighted graphs with a particular spectral property, identifying their structure as joins of unions of complete graphs, thus advancing understanding of graph spectra and structure.

Contribution

It provides a complete characterization of graphs with weighted adjacency matrices having a specified spectrum, linking spectral properties to graph construction.

Findings

Connected graphs are joins of unions of pairs of complete graphs.

Spectral characterization links graph structure to eigenvalues.

Identifies a class of graphs with specific spectral properties.

Abstract

In this paper we completely characterize the graphs which have an edge weighted adjacency matrix belonging to the class of $n \times n$ involutions with spectrum equal to $\{ \lambda_1^{n-2}, \lambda_2^{2} \}$ for some $\lambda_1$ and some $\lambda_2$. The connected graphs turn out to be the cographs constructed as the join of at least two unions of pairs of complete graphs, and possibly joined with one other complete graph.