The graphs which are cospectral with the generalized pineapple graph

TL;DR

This paper characterizes all graphs cospectral with the generalized pineapple graph $K_{p,k}^{q}$ by analyzing eigenvalues, extending previous results for specific cases to a broader class of graphs.

Contribution

It provides a complete spectral characterization of graphs cospectral with $K_{p,k}^{q}$, generalizing prior work for $k=1$ to all valid $k$ values.

Findings

All graphs cospectral with $K_{p,k}^{q}$ are determined.

Extension of previous cospectral graph results to broader parameters.

Eigenvalue analysis fully characterizes cospectrality for these graphs.

Abstract

Let $p, k, q$ be positive integers with $p-2 \geqslant k$ and let $K_{p,k}^{q}$ be the generalized pineapple graph which is obtained by joining independent set of $q$ vertices with $k$ vertices of a complete graph $K_{p}.$ In \cite{TSH2}, Haemers et al. constructed graphs which cospectral with $K_{p,1}^{q}.$ In this paper, we determine all graphs which are cospectral with $K_{p,k}^{q}$ by considering the eigenvalues of its adjacency matrix. Moreover, We extend the conclusions of Haemers et al. to a broader context.