On Color Energy of Few Classes of Bipartite Graphs and Corresponding Color Complements
Abstract:
For a given colored graph G, the color energy is defined as Ec(G) = Σλi, for i = 1, 2,…., n; where λi is a color eigenvalue of the color matrix of G, Ac (G) with entries as 1, if both the corresponding vertices are neighbors and have different colors; -1, if both the corresponding vertices are not neighbors and have same colors and 0, otherwise. In this article, we study color energy of graphs with proper coloring and L (h, k)-coloring. Further, we examine the relation between Ec(G) with the corresponding color complement of a given graph G and other graph parameters such as chromatic number and domination number.
AMS Subject Classification: 05C15, 05C50
Author(s):
DOI:
Keywords:
Color eigenvalues, color energy, color complement
References:
Adiga, C., Sampathkumar, E., Sriraj, M. A., Shrikanth, A. S.: Color Energy of a Graph. Proc. Jangjeon Math. Soc. 16, 335–351, (2013)
Adiga, C., Sampathkumar, E., Sriraj, M. A.: Color Energy of a Unitary Cayley Graph. Discuss. Math. Graph Theory. 4, 335–351, (2013) https://doi.org/10.7151/dmgt.1767
Betageri, K. S.: The Reduced Color Energy of Graphs. J. Comp. and Math. Sci. 7(1), 13-20, (2016)
Bhat, P. G., D’Souza, S.: Color Laplacian Energy of a Graph. Proc. Jangjeon Math. Soc. 18(3), 321–330,(2015)
Bhat, P. G., D’Souza, S.: Color Signless Laplacian Energy of Graphs. AKCE Int. J. of Graphs and Comb. 14(2), 142–148, (2017) https://doi.org/10.1016/j.akcej.2017.02.003
Chartrand, G., Zhang, P.: Coloring, distance and domination. In: Chromatic Graph Theory. CRC Press, New York (2008) https://doi.org/10.1201/9781584888017
Cvetkovic, D. M., Doob, M., Sachs, H.: Spectra of Graphs- Theory and Application, Academic Press, New York (1980)
Gutman, I.: The Energy of a Graph. Ber. Math. Stat. Sekt. Forschungsz. Graz.103, 1–22, (1978)
Gutman, I., Li, X., Zhang, J.: Graph Energy, Springer, New York (2012)
Gutman, I., Zhou, B.: Laplacian Energy of a Graph. Linear Algebra Appl. 414(1), 29–37, (2006) https://doi.org/10.1016/j.laa.2005.09.008
Hedetniemi S., Slater P., Haynes T. W. Fundamentals of Domination in Graphs, CRC press, (2013)
Indulal, G., Gutman, I., Ambat V.: On Distance Energy of Graphs. MATCH Comm. Math. Comp. Chem. 34(4), 461–472, (2010)
Jeyakokila, S. P., Sumathi, P.: A note on soEnergy of Stars, Bistars and Double stars graphs. Bull.Inter. Math. Virtual Inst. 6, 105-113, (2016)
Joshi, P. B., Joseph, M.: Further Results on Color Energy of Graphs. Acta Univ. Sapient., Info. 9, 191–131, (2017) https://doi.org/10.1515/ausi-2017-0008
Joshi, P. B., Joseph, M.: On New Bounds for Color Energy of Graphs. Int. J. Pure and Appl. Math. 117, 25–33, (2017)
Kanna, M. R., Kumar, R. P., Jagadeesh, R.: Minimum Covering Color Energy of a Graph. Asian Acad. Res.J. Multidiscip. 9 (8), 351–364, (2015) https://doi.org/10.12988/ijma.2015.412382
Sampath kumar, E., Sriraj, M. A.: Vertex Labeled/Colored Graphs, Matrices and Signed Graphs. J. Comb. Info. System Sci. 38, 113–120, (2013)
Shigehalli, V. S., Betageri, K. S.: Color Laplacian Energy of Graphs. J. Comp. and Math. Sci. 6(9), 485–494, (2015)
West, D. B.: Introduction to Graph Theory, Pearson,New Jersey (2001)
Adiga, C., Sampathkumar, E., Sriraj, M. A.: Color Energy of a Unitary Cayley Graph. Discuss. Math. Graph Theory. 4, 335–351, (2013) https://doi.org/10.7151/dmgt.1767
Betageri, K. S.: The Reduced Color Energy of Graphs. J. Comp. and Math. Sci. 7(1), 13-20, (2016)
Bhat, P. G., D’Souza, S.: Color Laplacian Energy of a Graph. Proc. Jangjeon Math. Soc. 18(3), 321–330,(2015)
Bhat, P. G., D’Souza, S.: Color Signless Laplacian Energy of Graphs. AKCE Int. J. of Graphs and Comb. 14(2), 142–148, (2017) https://doi.org/10.1016/j.akcej.2017.02.003
Chartrand, G., Zhang, P.: Coloring, distance and domination. In: Chromatic Graph Theory. CRC Press, New York (2008) https://doi.org/10.1201/9781584888017
Cvetkovic, D. M., Doob, M., Sachs, H.: Spectra of Graphs- Theory and Application, Academic Press, New York (1980)
Gutman, I.: The Energy of a Graph. Ber. Math. Stat. Sekt. Forschungsz. Graz.103, 1–22, (1978)
Gutman, I., Li, X., Zhang, J.: Graph Energy, Springer, New York (2012)
Gutman, I., Zhou, B.: Laplacian Energy of a Graph. Linear Algebra Appl. 414(1), 29–37, (2006) https://doi.org/10.1016/j.laa.2005.09.008
Hedetniemi S., Slater P., Haynes T. W. Fundamentals of Domination in Graphs, CRC press, (2013)
Indulal, G., Gutman, I., Ambat V.: On Distance Energy of Graphs. MATCH Comm. Math. Comp. Chem. 34(4), 461–472, (2010)
Jeyakokila, S. P., Sumathi, P.: A note on soEnergy of Stars, Bistars and Double stars graphs. Bull.Inter. Math. Virtual Inst. 6, 105-113, (2016)
Joshi, P. B., Joseph, M.: Further Results on Color Energy of Graphs. Acta Univ. Sapient., Info. 9, 191–131, (2017) https://doi.org/10.1515/ausi-2017-0008
Joshi, P. B., Joseph, M.: On New Bounds for Color Energy of Graphs. Int. J. Pure and Appl. Math. 117, 25–33, (2017)
Kanna, M. R., Kumar, R. P., Jagadeesh, R.: Minimum Covering Color Energy of a Graph. Asian Acad. Res.J. Multidiscip. 9 (8), 351–364, (2015) https://doi.org/10.12988/ijma.2015.412382
Sampath kumar, E., Sriraj, M. A.: Vertex Labeled/Colored Graphs, Matrices and Signed Graphs. J. Comb. Info. System Sci. 38, 113–120, (2013)
Shigehalli, V. S., Betageri, K. S.: Color Laplacian Energy of Graphs. J. Comp. and Math. Sci. 6(9), 485–494, (2015)
West, D. B.: Introduction to Graph Theory, Pearson,New Jersey (2001)
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