Graphs cospectral with a friendship graph or its complement

Abstract: Let $n$ be any positive integer and let $F_n$ be the friendship (or Dutch windmill) graph with $2n+1$ vertices and $3n$ edges. Here we study graphs with the same adjacency spectrum as the $F_n$. Two graphs are called cospectral if the eigenvalues multiset of their adjacency matrices are the same. Let $G$ be a graph cospectral with $F_n$. Here we prove that if $G$ has no cycle of length 4 or 5, then $G\cong F_n$. Moreover if $G$ is connected and planar then $G\cong F_n$. All but one of connected components of $G$ are isomorphic to $K_2$. The complement $\bar{F_n}$ of the friendship graph is determined by its adjacency eigenvalues, that is, if $\bar{F_n}$ is cospectral with a graph $H$, then $H\cong \bar{F_n}$.