Normalized laplacian spectrum of two new types of join graphs

Subject Areas : History and biography

M. Hakimi-Nezhaad ¹ , M. Ghorbani ²

1 - Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785-136, Iran
2 - Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785-136, Iran

Received: 2016-11-10 Accepted : 2017-04-20 Published : 2017-03-01

Keywords: Join of graphs‎, ‎normalized Laplacian eigenvalue, integral eigenvalue‎,

Abstract :

‎Let $G$ be a graph without an isolated vertex‎, ‎the normalized Laplacian matrix $\tilde{\mathcal{L}}(G)$‎‎is defined as $\tilde{\mathcal{L}}(G)=\mathcal{D}^{-\frac{1}{2}}\mathcal{L}(G)\mathcal{D}^{-\frac{1}{2}}$‎, where ‎$\mathcal{D}$ ‎is a‎ diagonal matrix whose entries are degree of ‎vertices ‎‎of ‎$‎G‎$‎‎. ‎The eigenvalues of‎$\tilde{\mathcal{L}}(G)$ are ‎called as ‎the ‎normalized Laplacian eigenvalues of $G$‎. ‎In this paper‎, ‎we obtain the normalized Laplacian spectrum of two new types of join graphs‎. ‎In continuing‎, ‎we determine the integrality of normalized Laplacian eigenvalues of graphs‎. ‎Finally‎, ‎the normalized Laplacian energy and degree Kirchhoff index of these new graph ‎products‎ are derived‎.

References:

[1] A. E. Brouwer, W. H. Haemers, Spectra of Graphs, Universitext, Springer, New York, 2012.

[2] S. Butler, Eigenvalues and Structures of Graphs, Ph.D. dissertation, University of California, San Diego, 2008.

[3] M. Cavers, The normalized Laplacian matrix and general Randic index of graphs, Ph.D. University of Regina, 2010.

[4] M. Cavers, S. Fallat, S. Kirkland, On the normalized Laplacian energy and general Randic index R-1 of graphs, Linear Algebra Appl. 433 (2010), 172-190.

[5] H. Chen, F. Zhang, Resistance distance and the normalized Laplacian spectrum, Discr. Appl. Math. 155 (2007), 654-661.

[6] F. R. K. Chung, Spectral Graph Theory, American Math. Soc. Providence, 1997.

[7] D. Cvetkovic, M. Doob, H. Sachs, Spectra of Graphs: Theory and Applications, Academic Press, New York, 1980.

[8] I. Gutman, The energy of a graph, Steiermrkisches Mathematisches Symposium (Stift Rein, Graz, 1978), Ber. Math. Statist. Sekt. Forsch. Graz 103 (1978) 1-22.

[9] I. Gutman, B. Mohar, The Quasi-Wiener and the Kirchho indices coincide, J. Chem. Inf. Comput. Sci. 36 (1996), 982-985.

[10] I. Gutman, The energy of a graph: old and new results, in: A. Betten, A. Kohner, R. Laue, A. Wassermann (Eds.), Algebraic Combinatorics and Applications, Springer, Berlin, 2001, 196-211.

[11] M. Hakimi-Nezhaad, A. R. Ashra, I. Gutman, Note on degree Kirchho index of graphs, Trans. Comb. 2 (3) (2013), 43-52.

[12] M. Hakimi-Nezhaad, A. R. Ashraf, A note on normalized Laplacian energy of graphs, J. Contemp. Math. Anal. 49 (5) (2014), 207-211.

[13] G. Indulal, Spectrum of two new joins of graphs and infinite families of integral graphs, Kragujevac J. Math. 36 (1) (2012), 133-139.

[14] E. R. Van Dam, G. R. Omidi, Graphs whose normalized Laplacian has three eigenvalues, Linear Algebra Appl. 435 (10) (2011), 2560-2569.

[15] M. R. Jooyandeh, D. Kiani, M. Mirzakhah, Incidence energy of a graph, MATCH Commun. Math. Comput. Chem. 62 (2009), 561-572.

[16] D. J. Klein, M. Randic, Resistance distance, J. Math. Chem. 12 (1993), 81-95.

[17] X. Li, Y. Shi, I. Gutman, Graph energy, Springer, New York, 2012.

[18] B. Zhou, N. Trinajstic, On resistance-distance and Kirchho index, J. Math. Chem. 46 (2009), 283-289.