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 $G\cong F_n$. Moreover if $G$ is connected and planar then $G\cong F_n$. All but one of connected components of $G$ are isomorphic to $K_2$. 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 $H\cong \overline{F_n}$.
Abdollahi, A., Janbaz, S., & Oboudi, M. R. (2013). Graphs cospectral with a friendship graph or its complement. Transactions on Combinatorics, 2(4), 37-52. doi: 10.22108/toc.2013.3621
MLA
Alireza Abdollahi; Shahrooz Janbaz; Mohammad Reza Oboudi. "Graphs cospectral with a friendship graph or its complement". Transactions on Combinatorics, 2, 4, 2013, 37-52. doi: 10.22108/toc.2013.3621
HARVARD
Abdollahi, A., Janbaz, S., Oboudi, M. R. (2013). 'Graphs cospectral with a friendship graph or its complement', Transactions on Combinatorics, 2(4), pp. 37-52. doi: 10.22108/toc.2013.3621
VANCOUVER
Abdollahi, A., Janbaz, S., Oboudi, M. R. Graphs cospectral with a friendship graph or its complement. Transactions on Combinatorics, 2013; 2(4): 37-52. doi: 10.22108/toc.2013.3621
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