Connected graphs cospectral with a Friendship graph

الأصل · EN

Let n be any positive integer, the friendship graph Fₙ 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 Fₙ (n≠ 16), then G Fₙ. In this note, we give a proof of special case of the latter: Any connected graph cospectral with Fₙ is isomorphic to Fₙ. 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.].