Degree square sum equienergetic and hyperenergetic graphs

B. Basavanagoud; E. Chitra

doi:10.26637/MJM0802/0001

Abstract

Degree square sum matrix $\operatorname{DSS}(G)$ of a graph $G$ is a square matrix of order equal to the order of a graph $G$ with its $(i, j)^{t h}$ entry as $d_i{ }^2+d_j{ }^2$ if $i \neq j$ and zero otherwise, where $d_i$ is the degree of the $i^{t h}$ vertex of $G$. In this paper, we study degree square sum hyperenergetic, degree square sum borderenergetic and degree square sum equienergetic graphs.

Keywords:

Degree square sum matrix, degree square sum polynomial, degree square sum energy, degree square sum hyperenergetic graphs, degree square sum equienergetic graphs

Mathematics Subject Classification:

Mathemativs

Authors Imprint References Funding Related Articles Downloads

B. Basavanagoud Department of Mathematics, Karnatak University, Dharwad-580003, Karnataka, India.
E. Chitra Department of Mathematics, Karnatak University, Dharwad-580003, Karnataka, India. https://orcid.org/0000-0002-7782-3422

Pages: 301-305
Date Published: 01-04-2020
Vol. 8 No. 02 (2020): Malaya Journal of Matematik (MJM)

R. Balakrishnan, The energy of a graph, Linear Algebra Appl., 387(2004), 287-295.

B. Basavanagoud, Chitra, E, Degree square sum energy of graphs, Int. J. Math. Appl., 6(2-B)(2018), 193-205.

B. Basavanagoud, Chitra, E, Degree square sum polynomial of some special graphs, Int. J. Appl. Eng. Res., 13(19)(2018), 14060-14078.

V. Brankov, D. Stevanovic, I. Gutman, Equienergetic chemical trees, J. Serb. Chem. Soc., 69(7)(2004), 549553.

A. E. Brouwer, W. H. Haemers, Spectra of graphs, Springer, Berlin, 2012.

F. Buckley, Iterated line graphs, Congr. Numer., 33(1981),390-394.

F. Buckley, The size of iterated line graphs, Graph Theory Notes New York, 25(1993), 33-36.

G. Chartrand, H. Hevia, E. B. Jarrett, M. Schultz, Subgraph distances in graphs defined by edge transfers, Discrete Math., 170(1997), 63-79.

D. Cvetković, M. Doob, H. Sachs, Spectra of graphsTheory and applications, Academic Press, New York, 1980.

D. Cvetković, I. Gutman, The computer system graph: A useful tool in chemical graph theory, J. Comput. Chem., 7(5)(1986), 640-644.

D. Cvetković, P. Rowlinson, S. K. Simić, Eigenvalue bounds for the signless Laplacian, Publ. Inst. Math., 81(95)(2007), 11-27.

S. Gong, X. Li, G. Xu, I. Gutman, B. Furtula, Borderenergetic graphs, MATCH Commun. Math. Comput. Chem., 74(2015), 321-332.

I. Gutman, The energy of a graph, Ber. Math. Stat. Sekt. Forschungsz. Graz, 103(1978), 1-22.

I Gutman, O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer-Verlag, Berlin, 1986.

I. Gutman, Hyperenergetic molecular graphs, J. Serbian Chem. Soc., 64(1999), 199-205.

I. Gutman, Open problems for equienergetic graphs, Iranian J. Math. Chem., 6(2015), 185-187.

F. Harary, Graph Theory, Addison-Wesely, Reading, Mass, 1969.

Y. Hou, I. Gutman, Hyperenergetic line graphs, MATCH Commun. Math. Comput. Chem., 43(2001), 29-39.

G. Indulal, A. Vijayakumar, On a pair of equienergetic graphs, MATCH Commun. Math. Comput. Chem., $55(2006), 83-90$.

G. Indulal, A. Vijayakumar, Energies of some nonregular graphs, J. Math. Chem., 42(3)(2007), 377-386.

J. H. Koolen, V. Moulton, Maximal energy graphs, $A d v$. Appl. Math., 26(2001), 47-52.

J. H. Koolen, V. Moulton, Maximal energy bipartite graphs, Graphs and Combin., 19(2003), 131-135.

V. R. Kulli, College Graph Theory, Vishwa Int. Publ., Gulbarga, India, 2012.

J. Liu, B. Liu, Note on a pair of equienergetic graphs, MATCH Commun. Math. Comput. Chem., 59(2008), 275 278.

X. Li, Y. Li, Y. Shi, Note on the energy of regular graphs, Linear Algebra Appl., 432(5)(2010), 1144-1146.

B. J. McClelland, Properties of the latent roots of a matrix: The estimation of $pi$-electron energies, J. Chem. Phy., 54(2)(1971), 640-643.

H. S. Ramane, I. Gutman, H. B. Walikar, S. B. Halkarni, Another class of equienergetic graphs, Kragujevac J. Math., 26(2004), 15-17.

H. S. Ramane, H. B. Walikar, Construction of equienergetic graphs, MATCH Commun. Math. Comput. Chem., 57(2007), 203-210.

H. S. Ramane, H. B. Walikar, I. Gutman, Equienergetic graphs, J. Combin. Math. Combin. Comput., 69(2009), $165-173$.

H. S. Ramane, H. B. Walikar, S. B. Rao, B. D. Acharya, P. R. Hampiholi, S. R. Jog, I. Gutman, Equienergetic graphs, Kragujevac J. Math., 26(2004), 5-13.

N. Trinajstić, Chemical Graph Theory, CRC Press, London, 1983.

H. B. Walikar, H. S. Ramane, S. R. Jog, On an open problem of R. Balakrishnan and the energy of products of graphs, Graph Theory Notes New York, 55(2008), 41-44.

L. Xu, Y. Hou, Equienergetic bipartite graphs, $M A T C H$ Commun. Math. Comput. Chem., 57(2007), 363-370.

B. Zhou, B, Energy of graphs, MATCH Commun. Math. Comput. Chem., 51(2004), 111-118.

B. Zhou, H. S. Ramane, On upper bounds for energy of bipartite graphs, Indian J. Pure Appl. Math., 39(2008), 483-490.