On the spectral characterization of Kite graphs

TL;DR

This paper investigates the spectral properties of Kite graphs, proving their spectral uniqueness and establishing bounds on graphs cospectral with them, thereby advancing understanding of their structural characterization.

Contribution

It demonstrates that no two non-isomorphic Kite graphs are cospectral and proves that Kite graphs with q=2 are uniquely determined by their adjacency spectrum.

Findings

No two non-isomorphic Kite graphs are cospectral.

Kite graphs with q=2 are determined by their adjacency spectrum.

A lower bound on the clique number of graphs cospectral with Kite graphs.

Abstract

The \textit{Kite graph}, denoted by $Kite_{p,q}$ is obtained by appending a complete graph $K_{p}$ to a pendant vertex of a path $P_{q}$. In this paper, firstly we show that no two non-isomorphic kite graphs are cospectral w.r.t adjacency matrix. Let $G$ be a graph which is cospectral with $Kite_{p,q}$ and the clique number of $G$ is denoted by $w(G)$. Then, it is shown that $w(G)\geq p-2q+1$. Also, we prove that $Kite_{p,2}$ graphs are determined by their adjacency spectrum.