GRAPHS COSPECTRAL WITH A FRIENDSHIP GRAPH OR ITS COMPLEMENT

Let n be any positive integer and Fn be the friendship (or Dutch windmill) graph with 2n+1 vertices and 3n edges. Here we study graphs with the same adjacency spectrum as Fn. Two graphs are called cospectral if the eigenvalues multiset of their adjacency matrices are the same. Let G be a graph cospectral with Fn. Here we prove that if G has no cycle of length 4 or 5, then G = Fn. Moreover if G is connected and planar then G = Fn. All but one of connected components of G are isomorphic to K2. The complement Fn of the friendship graph is determined by its adjacency eigenvalues, that is, if Fn is cospectral with a graph H, then H = Fn.