Energy of Cartesian product of Graphs
Keywords:
Cartesian Product, Adjacency Matrix, Energy of graphAbstract
An eigenvalue of a graph is an eigenvalue of its adjacency matrix. The energy of a graph is the sum of absolute values of its eigenvalues. Two graphs having same energy and same number of vertices are called equienergetic graphs. One might be interested to know, as to how the energy of a given graph can be related with the graph obtained from original graph by means of some graph operations. As an answer to this question we have considered the Cartesian product of two graphs. In this paper we obtain the eigenvalues and energy of Cartesian product of two graphs from the eigenvalue of the given graph.
References
[1] Andries E. Brouwer, Willem H. Haemers, “Spectra of Graph”, Monograph, Springer, February 1, 2011
[2] S. Avgustinovich and D. Fon-der-flaass, “Cartesian Products of Graphs and Metric Spaces”, Europ. J. Combinatorics, pp.847-851 (2000).
[3] R. Balakrishnan, “The Energy of a Graph”, Lin. Algebra Appl. 387 ,pp.287-295 (2004).
[4] R. Balakrishnan, K. Ranganathan, “A Textbook of Graph Theory”, Springer, New York, 2000.
[5] R. B. Bapat, S. Pati, “Energy of a graph is never an odd integer”, Bull. Kerala Math. Assoc.1 129-132 (2004).
[6] D. Cvetkovi_c, P. Rowlison, S. Simi_c, “An Introduction to the Theory of Graph Spectra”, Cambridge Univ. Press, Cambridge, 2010.
[7] Douglas B. West, “Introduction to Graph Theory”, University of Illinois, 2nd edition, 2001.
[8] I. Gutman, “The Energy of a Graph”, Ber. Math Statist. Sekt. Forschungsz. Graz 103,pp.1-22 (1978).
[9] I. Gutman, Y. Hou, H.B.Walikar, H.S. Ramane, P.R. Hampiholi, “No Hückel graph is Hyperenergetic”, J. Serb. Chem. Soc. 65 (11) 799–801 (2000).
[10] R. A. Horn, C. R. Johnson, “Topics in Matrix Analysis”, Cambridge Univ. Press, Cambridge, 1991
[11] S. Lang, “Algebra”, Springer, New York, 2002.
[12] X. Li, Y. Shi, I. Gutman, “Graph Energy”, Springer, New York, 2012.
[13] S. Pirzada, I. Gutman, “Energy of a graph is never the square root of an odd integer”, Appl. Anal. Discr. Math. 21, pp.18-121 (2008).
[14] Samir K. Vaidya and Kalpesh M. Popat, “Some new results on Energy of Graphs”, Match Commun. Math. Comput. Chem. 77, pp.589-594 (2017).
[15] M. A. Sriraj, “Some studies on energy of graphs, Ph. D. Thesis, Univ. Mysore, India, 2014.
[16] H.B. Walikar, I. Gutman, P.R. Hampiholi, H.S. Ramane, “Graph Theory Notes” New York Acad. Sci.41, pp.14–16 (2001).
[17] H.B.Walikar, H.S. Ramane, P.R. Hampiholi, “Energy of trees with edge independence number three”, preprint.
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