Graphs cospectral with a friendship graph or its complement

• Main Authors: Alireza Abdollahi, Shahrooz Janbaz, Mohammad Reza Oboudi
• Format: Article
• Language: English
• Published: University of Isfahan 2013-12-01
• Series: Transactions on Combinatorics
• Subjects: Friendship graphs; cospectral graphs; adjacency eigenvalues
• Online Access: http://www.combinatorics.ir/?_action=showPDF&article=3621&_ob=0bcd0f5df9a893e748683a0325f8cac6&fileName=full_text.pdf.

Let $n$ be any positive integer and let $F_n$ be the friendship (or Dutch windmill) graph with $2n+1$ vertices and $3n$ edges. Here we study graphs with the same adjacency spectrum as the $F_n$. 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 $Gcong F_n$. Moreover if $G$ is connected and planar then $Gcong F_n$.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 $Hcong overline{F_n}$.