A family of graphs that are determined by their normalized Laplacian spectra

TL;DR

This paper characterizes when generalized friendship graphs are uniquely identified by their normalized Laplacian spectra, showing they are determined by spectrum for most parameter values.

Contribution

It provides a complete characterization of the parameters for which generalized friendship graphs are uniquely determined by their normalized Laplacian spectra.

Findings

$F_{p,q}$ is spectrum-determined if and only if $q eq 1$ or $q=1$ and $p eq 3$.

The paper proves the spectrum-determined property for all $q eq 1$ and for $q=1$ with $p eq 3$.

It identifies the specific cases where the spectrum does not determine the graph.

Abstract

Let $F_{p,q}$ be the generalized friendship graph $K_1\bigvee (pK_q)$ on $pq+1$ vertices obtained by joining a vertex to all vertices of $p$ disjoint copies of the complete graph $K_q$ on $q$ vertices. In this paper, we prove that $F_{p,q}$ is determined by its normalized Laplacian spectrum if and only if $q\ge 2$, or $q=1$ and $p\le 2$.