@article {
author = {Abdollahi, Alireza and Janbaz, Shahrooz and Oboudi, Mohammad Reza},
title = {Graphs cospectral with a friendship graph or its complement},
journal = {Transactions on Combinatorics},
volume = {2},
number = {4},
pages = {37-52},
year = {2013},
publisher = {University of Isfahan},
issn = {2251-8657},
eissn = {2251-8665},
doi = {10.22108/toc.2013.3621},
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 $G\cong F_n$. 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 $H\cong \overline{F_n}$.},
keywords = {Friendship graphs,cospectral graphs,adjacency eigenvalues},
url = {https://toc.ui.ac.ir/article_3621.html},
eprint = {https://toc.ui.ac.ir/article_3621_0bcd0f5df9a893e748683a0325f8cac6.pdf}
}