Spectral characterization of multicone graphs

A multicone graph is defined to be the join of a clique and a regular graph. Based on Zhou and Cho’s result [B. Zhou, H.H. Cho, Remarks on spectral radius and Laplacian eigenvalues of a graph, Czech. Math. J. 55 (130) (2005), 781–790], the spectral characterization of multicone graphs is investigated. Particularly, we determine a necessary and sufficient condition for two multicone graphs to be cospectral graphs and investigate the structures of graphs cospectral to a multicone graph. Additionally, lower and upper bounds for the largest eigenvalue of a multicone graph are given.

[1]  H. Günthard,et al.  Zusammenhang von Graphentheorie und MO-Theorie von Molekeln mit Systemen konjugierter Bindungen , 1956 .

[2]  A. Rényii,et al.  ON A PROBLEM OF GRAPH THEORY , 1966 .

[3]  Frank Harary,et al.  Cospectral Graphs and Digraphs , 1971 .

[4]  J. Q. Longyear,et al.  The friendship theorem , 1972 .

[5]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[6]  Charles R. Johnson,et al.  A note on cospectral graphs , 1980, J. Comb. Theory, Ser. B.

[7]  Michael Doob,et al.  Spectra of graphs , 1980 .

[8]  B. McKay,et al.  Constructing cospectral graphs , 1982 .

[9]  Yuan Hong,et al.  A Sharp Upper Bound of the Spectral Radius of Graphs , 2001, J. Comb. Theory, Ser. B.

[10]  Vladimir Nikiforov,et al.  Some Inequalities for the Largest Eigenvalue of a Graph , 2002, Combinatorics, Probability and Computing.

[11]  Craig Huneke.  The Friendship Theorem , 2002, Am. Math. Mon..

[12]  W. Haemers,et al.  Which graphs are determined by their spectrum , 2003 .

[13]  Willem H. Haemers,et al.  Enumeration of cospectral graphs , 2004, Eur. J. Comb..

[14]  Han Hyuk Cho,et al.  Remarks on Spectral Radius and Laplacian Eigenvalues of a Graph , 2005 .

[15]  W. Haemers,et al.  Developments on Spectral Characterizations of Graphs , 2007 .

[16]  Willem H. Haemers,et al.  Developments on Spectral Characterizations of Graphs , 2007, Discret. Math..

[17]  Jianfeng Wang,et al.  On the two largest Q-eigenvalues of graphs , 2010, Discret. Math..