CONNECTED GRAPHS COSPECTRAL WITH A FRIENDSHIP GRAPH

Abstract

Let n be any positive integer, the friendship graph F, 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 http://arxiv.org/pdf/1310.6529v1.pdf it is proved that if G is any graph cospectral with F, (n not equal 16), then G similar or equal to F-n. Here we give a proof of a special case of the latter: Any connected graph cospectral with F, 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. Comb., 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.].