GRAPHS COSPECTRAL WITH A FRIENDSHIP GRAPH OR ITS COMPLEMENT

Abstract

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