Connected graphs cospectral with a friendship graph

‎Let 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/۱۳۱۰.۶۵۲۹v۱.pdf}{http://arxiv.org /pdf/۱۳۱۰.۶۵۲۹v۱.pdf} it is proved that if G is any graph cospectral with F_n (n\neq ۱۶)‎, ‎then G\cong 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/۱۳۱۰.۶۵۲۹v۱.pdf}{http://arxiv.org/pdf/۱۳۱۰.۶۵۲۹v۱.pdf} and the proofs are based on our recent results given in [Trans‎. ‎Comb.‎, ‎ ۲ no‎. ‎۴ (۲۰۱۳) ۳۷-۵۲.] using an upper bound for the largest eigenvalue of a connected graph given in‎ ‎[J‎. ‎Combinatorial Theory Ser‎. ‎B, ۸۱ (۲۰۰۱) ۱۷۷-۱۸۳.]‎.

Keywords:

Friendship graphs , cospectral graphs , adjacency eigenvalues , spectral radius