Graphs cospectral with a friendship graph or its complement
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. Le...
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doaj-00600a682e01418394a3e3858939592d2020-11-24T22:11:38ZengUniversity of IsfahanTransactions on Combinatorics2251-86572251-86652013-12-01243752Graphs cospectral with a friendship graph or its complementAlireza AbdollahiShahrooz JanbazMohammad Reza OboudiLet $n$ 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 $Hcong overline{F_n}$.http://www.combinatorics.ir/?_action=showPDF&article=3621&_ob=0bcd0f5df9a893e748683a0325f8cac6&fileName=full_text.pdf.Friendship graphscospectral graphsadjacency eigenvalues |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Alireza Abdollahi Shahrooz Janbaz Mohammad Reza Oboudi |
spellingShingle |
Alireza Abdollahi Shahrooz Janbaz Mohammad Reza Oboudi Graphs cospectral with a friendship graph or its complement Transactions on Combinatorics Friendship graphs cospectral graphs adjacency eigenvalues |
author_facet |
Alireza Abdollahi Shahrooz Janbaz Mohammad Reza Oboudi |
author_sort |
Alireza Abdollahi |
title |
Graphs cospectral with a friendship graph or its complement |
title_short |
Graphs cospectral with a friendship graph or its complement |
title_full |
Graphs cospectral with a friendship graph or its complement |
title_fullStr |
Graphs cospectral with a friendship graph or its complement |
title_full_unstemmed |
Graphs cospectral with a friendship graph or its complement |
title_sort |
graphs cospectral with a friendship graph or its complement |
publisher |
University of Isfahan |
series |
Transactions on Combinatorics |
issn |
2251-8657 2251-8665 |
publishDate |
2013-12-01 |
description |
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 $Hcong overline{F_n}$. |
topic |
Friendship graphs cospectral graphs adjacency eigenvalues |
url |
http://www.combinatorics.ir/?_action=showPDF&article=3621&_ob=0bcd0f5df9a893e748683a0325f8cac6&fileName=full_text.pdf. |
work_keys_str_mv |
AT alirezaabdollahi graphscospectralwithafriendshipgraphoritscomplement AT shahroozjanbaz graphscospectralwithafriendshipgraphoritscomplement AT mohammadrezaoboudi graphscospectralwithafriendshipgraphoritscomplement |
_version_ |
1725804808499101696 |