On the spectral characterization of mixed extensions of $P_3$

TL;DR

This paper investigates the spectral properties of mixed extensions of the path graph $P_3$, classifies those determined by their spectra, and identifies cospectral families through computational methods.

Contribution

It classifies mixed extensions of $P_3$ that are uniquely determined by their adjacency spectra and provides a comprehensive list of cospectral graphs up to 25 vertices.

Findings

Identified all graphs cospectral with mixed extensions of $P_3$ up to 25 vertices.

Classified mixed extensions of $P_3$ based on their spectral properties.

Presented several cospectral families of graphs.

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, whilst two vertices in $H$ corresponding to distinct vertices $x$ and $y$ of $G$ are adjacent whenever $x$ and $y$ are adjacent in $G$. If $G$ is the path $P_3$, then $H$ has at most three adjacency eigenvalues unequal to $0$ and $-1$. Recently, the first author classified the graphs with the mentioned eigenvalue property. Using this classification we investigate mixed extension of $P_3$ on being determined by the adjacency spectrum. We present several cospectral families, and with the help of a computer we find all graphs on at most $25$ vertices that are cospectral with a mixed extension of $P_3$.