Let $n\geq 1$ be an integer. The Friendship Graph (or Dutch windmill graph or $n$-Fan) $F\_n$ is a graph that can be constructed by coalescence $n$ copies of the cycle graph $C\_3$ with a common vertex. By construction, the friendship graph $F\_n$ is isomorphic to the windmill graph $Wd\left(3,n\right)$.


Can $F\_n$ be determined by its adjacency spectrum?
By the adjacency spectrum of a graph, we mean the multiset of the eigenvalues of the adjacency matrix of the graph. For a graph $G$, we denote by $Spec(G)$ its adjacency spectrum.
A graph $G$ is said to be determined by its adjacency spectrum, if $Spec(G)=Spec(H)$ for some graph $H$, then $G\cong H$. 


It is known that the friendship graph can be determined by the signless Laplacian spectrum.
See [Discrete Math. 310, No. 21, 2858-2866 (2010).]



Can the friendship graph be determind by its adjacency spectrum?
This isn't a real answer, but wolfram-alpha says that $F\_n$ is determined by its (adjacency) spectrum for $n \in \lbrace 2,3,4 \rbrace $. It doesn't say anything about $n=5$.


<http://www.wolframalpha.com/input/?i=%285%2C3%29-windmill+graph>


