# Graphs cospectral with a friendship graph or its complement

Transactions on Combinatorics. 2013;2(4):37-52

**Journal Title**: Transactions on Combinatorics

**ISSN**:
2251-8657 (Print); 2251-8665 (Online)

**Publisher**: University of Isfahan

**LCC Subject Category**:
Science: Mathematics

**Country of publisher**: Iran, Islamic Republic of

**Language of fulltext**: English

**Full-text formats available**: PDF

**AUTHORS**

*Alireza Abdollahi
*

*Shahrooz Janbaz
*

*Mohammad Reza Oboudi
*

**EDITORIAL INFORMATION**

Time From Submission to Publication: 30 weeks

**
Abstract
| Full Text
**

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 $Hcong overline{F_n}$.