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 $F_n$ ($n\neq 16$), then $G\cong F_n$. Here 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 [Trans. Comb., 2 no. 4 (2013) 37-52.] using an upper bound for the largest eigenvalue of a connected graph given in [J. Combinatorial Theory Ser. B,81 (2001) 177-183.].
A. Abdollahi, S. Janbaz and M. R. Oboudi (2013). Graphs cospectral with a friendship graph or its complement. Trans. Comb.. 2 (4), 37-52 S. M. Cioaba, W. H. Haemers, J. Vermette and W. Wong (2013). The graphs with all but two eigenvalues equal to 1. http://arxiv.org/pdf/1310.6529v1.pdf. K. C. Das (2013). Proof of conjectures on adjacency eigenvalues of graphs. Discrete Math.. 313, 19-25 Y. Hong, J. Shu and K. Fang (2001). A sharp upper bound of the spectral radius of graphs. J. Combin. Theory Ser. B. 81, 177-183 J. F. Wang, F. Belardo, Q. X. Huang and B. Borovicanin (2010). On the two largest Q-eigenvalues of graphs. Discrete Math.. 310, 2858-2866 J. F. Wang, H. Zhao and Q. Huang (2012). Spectral characterization of multicone graphs. Czech. Math. J.. 62 (137), 117-126
Abdollahi, A. and Janbaz, S. (2014). Connected graphs cospectral with a friendship graph. Transactions on Combinatorics, 3(2), 17-20. doi: 10.22108/toc.2014.4975
MLA
Abdollahi, A. , and Janbaz, S. . "Connected graphs cospectral with a friendship graph", Transactions on Combinatorics, 3, 2, 2014, 17-20. doi: 10.22108/toc.2014.4975
HARVARD
Abdollahi, A., Janbaz, S. (2014). 'Connected graphs cospectral with a friendship graph', Transactions on Combinatorics, 3(2), pp. 17-20. doi: 10.22108/toc.2014.4975
CHICAGO
A. Abdollahi and S. Janbaz, "Connected graphs cospectral with a friendship graph," Transactions on Combinatorics, 3 2 (2014): 17-20, doi: 10.22108/toc.2014.4975
VANCOUVER
Abdollahi, A., Janbaz, S. Connected graphs cospectral with a friendship graph. Transactions on Combinatorics, 2014; 3(2): 17-20. doi: 10.22108/toc.2014.4975
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