graphs cospectral with a friendship graph or its complement

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 $gcong f_n$. moreover if $g$ is connected and planar then $gcong f_n$.all but one of connected components of $g$ are isomorphic to $k_2$.the complement $overline{f_n}$ of the friendship graph is determined by its adjacency eigenvalues, that is, if $overline{f_n}$ is cospectral with a graph $h$, then $hcong overline{f_n}$.