Connected graphs cospectral with a Friendship graph

Abstract

Let $n$ be any positive integer, the friendship graph $F_n$ consist 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 [http://arxiv.org/pdf/1310.6529v1.pdf] it is proved that if $G$ is any graph cospectral with $F_n$ $(n\neq 16)$, then $G\cong F_n$. In this note, we give a proof of 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 [http://arxiv.org/pdf/1310.6529v1.pdf] and the proofs are based on our recent results given in [Trans. Com., 2 no. 4 (2013) 37-52.] Using an upper bound for the largest eigenvalue of a connected graph given in [J. Combinatorial Theory, Ser. B, 81 (2001) 177-183.].