University of IsfahanTransactions on Combinatorics2251-86573220140601Connected graphs cospectral with a friendship graph1720497510.22108/toc.2014.4975ENAlirezaAbdollahiUniversity of IsfahanShahroozJanbazUniversity of IsfahanJournal Article20140228Let $n$ be any positive integer, the friendship graph $F_n$ consists of $n$ edge-disjoint triangles that all of them meeting in one vertex. A graph $G$ is called cospectral with a graph $H$ if their adjacency matrices have the same eigenvalues. Recently in href{http://arxiv.org/pdf/1310.6529v1.pdf}{http://arxiv.org /pdf/1310.6529v1.pdf} it is proved that if $G$ is any graph cospectral with $F_n$ ($nneq 16$), then $Gcong F_n$. Here we give a proof of a special case of the latter: Any connected graph cospectral with $F_n$ is isomorphic to $F_n$. Our proof is independent of ones given in href{http://arxiv.org/pdf/1310.6529v1.pdf}{http://arxiv.org/pdf/1310.6529v1.pdf} and the proofs are based on our recent results given in [<em>Trans. Comb.</em>, <strong>2</strong> no. 4 (2013) 37-52.] using an upper bound for the largest eigenvalue of a connected graph given in [<em>J. Combinatorial Theory Ser. B,</em> <strong>81</strong> (2001) 177-183.].https://toc.ui.ac.ir/article_4975_b084afb2f80121996ddeade80cc392f1.pdf