%0 Journal Article
%T Graphs cospectral with a friendship graph or its complement
%J Transactions on Combinatorics
%I University of Isfahan
%Z 2251-8657
%A Abdollahi, Alireza
%A Janbaz, Shahrooz
%A Oboudi, Mohammad Reza
%D 2013
%\ 12/01/2013
%V 2
%N 4
%P 37-52
%! Graphs cospectral with a friendship graph or its complement
%K Friendship graphs
%K cospectral graphs
%K adjacency eigenvalues
%R 10.22108/toc.2013.3621
%X Let $n$ 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 $Hcong overline{F_n}$.
%U https://toc.ui.ac.ir/article_3621_0bcd0f5df9a893e748683a0325f8cac6.pdf