### Degree Sum Exponent Distance Energy of Some Graphs

Jeetendra Gurjar, Sudhir Raghunath Jog

#### Abstract

The degree sum exponent distance matrix M(G)of a graph G is a square matrix whose (i,j)-th entry is (di+dj)^ d(ij) whenever i not equal to j, otherwise it is zero, where di is the degree of i-th vertex of G and d(ij)=d(vi,vj) is distance between vi and vj. In this paper, we define degree sum exponent distance energy E(G) as sum of absolute eigenvalues of M(G). Also, we obtain some bounds on the degree sum exponent distance energy of some graphs and deduce direct  expressions for some graphs.

#### Keywords

Degree sum exponent distance matrix, Degree sum exponent distance energy.

PDF

#### References

Gutman, I., "The Energy of a Graph", Ber. Math. Stat. Sekt. Forschungsz. Graz , 103 (1978), pp.1-22.

Gutman, I., Zhou, B., "Laplacian energy of a graph", Linear Algebra and its appl., 414(1), (2006), 29-37.

Cvetkovic, D., Rowlinson, P., and Simic, S., "Signless Laplacians of finite graphs", Linear Algebra and its Appl. 423(1) (2007), 155-171.

Das, K.Ch. and Mojallal, S. Ahhad , "On energy and Laplacian energy of graphs", Elec. J. of Linear Algebra, 31 (2016), 167-186.

Indulal, G., Gutman, I., and Vijaykumar, A.," On distance energy of graphs", MATCH Commun. Math. Comput. Chem 60 (2008), 461-472.

Jog, S.R. and Kotambari, R., "Degree Sum Energy of Some Graphs ", Annals of Pure and Appl. Math.11(1) (2016),17-27.

Meenakshi, S. and Lavanya, S., "A Survey on Energy of Graphs", Annals of Pure and Appl. Math.8(2) (2014), 183-191.

Basavanagoud, B. and Chitra, E., "Degree Square Sum Polynomial of some Special Graphs", Int. J. Math. And Appl.13(19) (2018), 14060-14078.

Ramane, H.S., Revenkar, D.S., and Patil, J.B., "Bounds for the degree sum eigenvalues and degree sum energy of a graph",Int. J. Pure App. Sci, 6(2) (2013), 161-167.

Ramane, H.S. and Nandesh, K.C., "Complementary distance spectra and complementary distance energy of line graph of regular graphs",J. Indones. Math. Soc 22(1) (2016), 2735.

Jog, S.R. and Gurjar, J.R., "Degree sum distance energy of some graphs", communicated.

Jog, S.R. and Gurjar, J.R., "Degree product distance energy of some graphs", Asian J. Math and Comp.Res. 24(1) (2018) 42-49.

Ramane, H.S. and Shinde, S.S., "Degree exponent polynomial and degree exponent energy of graphs", Indian J. Discrete Math 2(1), (2016), 1-7.

Ramane, H.S. and Shinde, S.S., "Degree exponent polynomial of graphs obtained by some graph operations", Electronics Notes in Discrete Mathematics 63 (2017), 161-168.

Kober, H., "On the arithmetic and geometric means and on Holder’s inequality".Proc. Amer. Math. Soc. 9 (1958), 452-459.

Basavanagoud, B. and Eshwarachandra, C., "Degree exponent sum energy of a graph", Gulf J. of math. 8(1) (2020), 52-70.

Grantmacher, F.R., Theory of Matrices, I Chelsea New York, 1960.

Davis, P.J., Circulant Matrices, Wiley, New York, 1979.

Bapat, R.B., Graphs and matrices, Springer, New York, 2010.

DOI: https://doi.org/10.22342/jims.27.1.931.64-74

### Refbacks

• There are currently no refbacks.

Journal of the Indonesian Mathematical Society