Graphs cospectral with a friendship graph or its complement

‎Let n be any positive integer and F_n be the friendship (or Dutch windmill) graph with ۲n+۱ vertices and ۳n 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 ۴ or ۵‎, ‎then G\cong F_n‎. ‎Moreover if G is connected and planar then G\cong F_n‎. ‎All but one of connected components of G are isomorphic to K_۲‎. ‎The complement \overline{F_n} of the friendship graph is determined by its adjacency eigenvalues‎, ‎that is‎, ‎if \overline{F_n} is cospectral with a graph H‎, ‎then H\cong \overline{F_n}‎.

Keywords:

Friendship graphs , cospectral graphs , adjacency eigenvalues