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Analytical methods to obtain the Energy of graphs

Abstract --

The energy of a graph is the sum of the absolute values of the eigenvalues of the adjacency matrix of the graph in consideration. This quantity is studied in the context of spectral graph theory,and is related to the sum of -electron energy in a molecule represented by a molecular graph- i.e. a graph where the vertices represent atoms and the edges bonds between atoms. Finding a simple expression for the energy of a class of graphs is therefore necessary and can be trivial, especially if the eigenvalues are integral (for the complete graph, for example). However it can be a challenge for other graphs, for example, for the cycle and path graphs  on  vertices, the energy of the graph involves the sum of  terms of cosine of a rational function of 1/n. In this paper, we use analytical methods to express the energy of cycles, paths and wheels, on  vertices, in terms of simplified expressions involving an isolated term of either cotangent or secant of a function of . This allowed us to show that these classes of graphs have the same energy of  for large values of , confirming that they are nothyperenergetic graphs.

Indexing Terms/Keyword --

Eigenvalues, energy of graphs, hypoenergetic graphs.

References --

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 [13]      Winter, P. A. and Jessop, C.L. 2014. Integral eigen-pair balanced classes of graphs with their ratio, asymptote, area and involution complementary aspects. International Journal of Combinatorics.Volume 2014.Article ID 148690, 16 pages.

 [14]      Winter, P. A. and Jessop, C.L. 2015. The Eigen-Cover Ratio of Graphs: Asymptotes, Domination and Areas. Global Journal of Mathematics.Volume 2, Issue 2, 37-47.

 [15]      Winter, P. A., Jessop, C. L. and Adewusi, F. J. 2015. The complete graph: eigenvalues, trigonometrical unit-equations with associated t-complete-eigen sequences, ratios, sums and diagrams. To appear in Journal of Advances in Mathematics.

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About the Author

Carol Jessop

Carol Jessop

Phd student, University of KwaZulu Natal, South Africa

jessop@telkomsa.net



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