Connected graphs cospectral with a Friendship graph

• Main Authors: Alireza Abdollahi, Shahrooz Janbaz
• Format: Article
• Language: English
• Published: University of Isfahan 2014-06-01
• Series: Transactions on Combinatorics
• Subjects: Friendship graphs; cospectral graphs; adjacency eigenvalues; spectral radius
• Online Access: http://www.combinatorics.ir/pdf_4975_b084afb2f80121996ddeade80cc392f1.html

Let $n$ be any positive integer, the friendship graph $F_n$ consists of $n$ edge-disjoint triangles that all of them meeting in one vertex. A graph $G$ is called cospectral with a graph $H$ if their adjacency matrices have the same eigenvalues. Recently in href{http://arxiv.org/pdf/1310.6529v1.pdf}{http://arxiv.org/pdf/1310.6529v1.pdf} it is proved that if $G$ is any graph cospectral with $F_n$ ($nneq 16$), then $Gcong F_n$. In this note, we give a proof of a special case of the latter: Any connected graph cospectral with $F_n$ is isomorphic to $F_n$. Our proof is independent of ones given in href{http://arxiv.org/pdf/1310.6529v1.pdf}{http://arxiv.org/pdf/1310.6529v1.pdf} and the proofs are based on our recent results given in [{em Trans. Comb.}, {bf 2} no. 4 (2013) 37-52.] using an upper bound for the largest eigenvalue of a connected graph given in [{em J. Combinatorial Theory Ser. B} {bf 81} (2001) 177-183.].