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Connected graphs cospectral with a friendship graph | ||
| Transactions on Combinatorics | ||
| مقاله 3، دوره 3، شماره 2، شهریور 2014، صفحه 17-20 اصل مقاله (108.72 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/toc.2014.4975 | ||
| نویسندگان | ||
| Alireza Abdollahi* ؛ Shahrooz Janbaz | ||
| University of Isfahan | ||
| چکیده | ||
| 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.]. | ||
| کلیدواژهها | ||
| Friendship graphs؛ cospectral graphs؛ adjacency eigenvalues؛ spectral radius | ||
| مراجع | ||
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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
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