Color Energy of Some Cluster Graphs

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/x5522-9720-4842-z Color Energy of Some Cluster Graphs D'Souza, S. , Girija, K. P. , Gowtham, H. J. , P. G. Bhat Vladikavkaz Mathematical Journal 2021. Vol. 23. Issue 2. Abstract: Let \(G\) be a simple connected graph. The energy of a graph \(G\) is defined as sum of the absolute eigenvalues of an adjacency matrix of the graph \(G\). It represents a proper generalization of a formula valid for the total \(\pi\)-electron energy of a conjugated hydrocarbon as calculated by the Huckel molecular orbital (HMO) method in quantum chemistry. A coloring of a graph \(G\) is a coloring of its vertices such that no two adjacent vertices share the same color. The minimum number of colors needed for the coloring of a graph \(G\) is called the chromatic number of \(G\) and is denoted by \(\chi(G)\). The color energy of a graph \(G\) is defined as the sum of absolute values of the color eigenvalues of \(G\). The graphs with large number of edges are referred as cluster graphs. Cluster graphs are graphs obtained from complete graphs by deleting few edges according to some criteria. It can be obtained on deleting some edges incident on a vertex, deletion of independent edges/triangles/cliques/path P3 etc. Bipartite cluster graphs are obtained by deleting few edges from complete bipartite graphs according to some rule. In this paper, the color energy of cluster graphs and bipartite cluster graphs are studied. Keywords: color adjacency matrix, color eigenvalues, color energy Language: English Download the full text For citation: D'Souza, S., Girija, K. P., Gowtham, H. J. and Bhat, P. G. Color Energy of Some Cluster Graphs,Vladikavkaz Math. J., 2021, vol. 23, no. 2, pp. 51-64. DOI 10.46698/x5522-9720-4842-z + References 1. Gutman, I. The Energy of a Graph, Ber. Math. Stat. Sekt. Forschungsz. Graz., 1978, vol. 103, pp. 1-22. 2. Harary, F. Graph Theory, New Delhi, Narosa Publishing House, 1989. 3. Adiga, C., Sampathkumar, E., Sriraj, M. A. and Shrikanth, A. S. Color Energy of a Graph, Proceedings of the Jangjeon Mathematical Society, 2013, vol. 16, pp. 335-351. 4. Abreu, N. M. M., Vinagre, C. T. M., Bonifacio, A. S. and Gutman, I. The Laplacian Energy of Some Laplacian Integral Graphs, MATCH Communications in Mathematical and in Computer Chemistry, 2008, vol. 60, no. 2, pp. 447-460. 5. Adiga, C., Sampathkumar, E. and Sriraj, M. A. Color Energy of Unitary Cayley Graphs, Discussiones Mathematicae Graph Theory, 2014, vol. 34, no. 4, pp. 707-721. DOI: 10.7151/dmgt.1767. 6. Balakrishnan, R. The Energy of a Graph, Linear Algebra and its Applications, 2004, vol. 387, pp. 287-295. DOI: 10.1016/j.laa.2004.02.038. 7. Balakrishnan, R. B. Graphs and Matrices, London, Springer-Hindustan Book Agency, 2011. 8. Bapat, R. B. and Pati, S. Energy of a Graph Is Never an Odd Integer, Bulletin of Kerala Mathematical Association, 2014, vol. 1, pp. 129-132. 9. Bhat, P. G. and D'souza, S. Color Laplacian Energy of a Graph, Proceedings of the Jangjeon Mathematical Society, 2015, vol. 18, pp. 321-330. 10. Bhat, P. G. and D'souza, S. Color Signless Laplacian Energy of a Graph, AKCE International Journal of Graphs and Combinatorics, 2017, no. 2, vol. 14, pp. 142-148. DOI: 10.1016/j.akcej.2017.02.003. 11. Gutman, I. and Pavlovic, L. The Energy of Some Graphs with Large Number of Edges, Bulletin of the Serbian Academy of Sciences. Mathematical and Natural Sciences Class, 1999, vol. 118, pp. 35-50. 12. Walikar, H. B. and Ramane, H. S. Energy of Some Bipartite Cluster Graphs, Kragujevac Journal of Science, 2001, vol. 23, pp. 63-74. 13. Walikar, H. B. and Ramane, H. S. Energy of Some Cluster Graphs, Kragujevac Journal of Science, 2001, vol. 23, pp. 51-62. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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