On the spectral characterization of pineapple graphs

TL;DR

This paper investigates the spectral properties of pineapple graphs, disproving a previous claim that they are uniquely determined by their spectra, except for specific cases where the claim holds.

Contribution

The authors construct counterexamples showing pineapple graphs are not spectrally unique for most parameters, and prove the claim holds only when q=2.

Findings

Counterexamples for p≥4 and various q

Proof that pineapple graphs are spectrally unique when q=2

References to literature for special cases

Abstract

The pineapple graph $K_p^q$ is obtained by appending $q$ pendant edges to a vertex of a complete graph $K_{p}$ ($q\geq 1,\ p\geq 3$). Zhang and Zhang ["Some graphs determined by their spectra", Linear Algebra and its Applications, 431 (2009) 1443-1454] claim that the pineapple graphs are determined by their adjacency spectrum. We show that their claim is false by constructing graphs which are cospectral and non-isomorphic with $K_p^q$ for every $p\geq 4$ and various values of $q$. In addition we prove that the claim is true if $q=2$, and refer to the literature for $q=1$, $p=3$, and $(p,q)=(4,3)$.