University of IsfahanTransactions on Combinatorics2251-86572420131201Graphs cospectral with a friendship graph or its complement3752362110.22108/toc.2013.3621ENAlireza AbdollahiUniversity of IsfahanShahrooz JanbazUniversity of IsfahanMohammad Reza OboudiUniversity of IsfahanJournal Article20130720Let $n$ be any positive integer and $F_n$ be the friendship (or Dutch windmill) graph with $2n+1$ vertices and $3n$ edges. Here we study graphs with the same adjacency spectrum as $F_n$. 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 $Hcong overline{F_n}$.http://toc.ui.ac.ir/article_3621_0bcd0f5df9a893e748683a0325f8cac6.pdf