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 --
[1] Brouwer, A. E. Haemers, W. H. Spectra of Graphs. Springer, New York. (2011).
[2] Chris Monica, M. and Santhakumar, S. Energy of Certain Planar Graphs. International Journal of Computing Algorithm Integrated Intelligent Research (IIR) Volume 03, February 2014 Pages: 554-557
[3] Coulson, C. A. O'Leary, B and Mallion, R. B.1978. Hückel Theory for Organic Chemists.Academic Press.
[4] Gutman, I.The energy of a graph, 10.SteiermarkischesMathematisches Symposium (Stift Rein, Graz, 1978), 103 (1978) 1-22.
[5] Gutman, I. Soldatovi´c, T. and Vidovi´c, D. The energy of a graph and its size dependence.A Monte Carlo approach, Chem. Phys. Lett. 297 (1998), 428–432.
[6] Haemers, W. H., Liu, X. and Zhang, Y. 2008. Spectral characterizations of lollipop graphs.Linear Algebra and its Applications. Volume 428, Issue 11–12, 2415–2423.
[7] Harris, J. M., Hirst, J. L. and Mossinghoff, M. 2008. Combinatorics and Graph theory.Springer, New York.
[8] Indulal, G. and Vijayakumar, A. A note on energy of some graphs.Communications in Mathematical and in Computer Chemistry 59 (2008), 269-274.
[9] Jessop, C. L. 2014. Matrices of Graphs and Designs with Emphasis on their Eigen-Pair Balanced Characteristic. M. Sc. Dissertation, University of Kwa-Zulu Natal.
[10] Koolen, Moulton, Gutman, Vidovi´c, More hyperenergetic molecular graphs. J. Serb. Chem. Soc. 65 (8) (2000), 571–575.
[11] Rojo, O. Line graph eigenvalues and line energy of caterpillars.Linear Algebra and its Applications 435 (2011) 2077-2086.
[12] Stevanovi´c, D. Energy of Graphs. A few open problems and some suggestions. Dragan Stevanovi´c dragance106@yahoo.com. University of Niˇs, Serbia.
[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.
[16] Woods, C My Favorite Application Using Graph Eigenvalues: Graph Energy. February 27, 2013
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About the Author
Carol Jessop Carol Jessop Phd student, University of KwaZulu Natal, South Africa |