Randic incidence energy of graphs

Document Type : Research Paper


1 Nankai University

2 Center for Combinatorics, Nankai University, Tianjin 300071, China


Let $G$ be a simple graph with vertex set $V(G) = \{v_1, v_2,\ldots, v_n\}$ and edge set $E(G) = \{e_1, e_2,\ldots, e_m\}$. Similar to the Randi'c matrix, here we introduce the Randi'c incidence matrix of a graph $G$, denoted by $I_R(G)$, which is defined as the $n\times m$ matrix whose $(i,j)$-entry is $(d_i)^{-\frac{1}{2}}$ if $v_i$ is incident to $e_j$ and $0$ otherwise. Naturally, the Randi'c incidence energy $I_RE$ of $G$ is the sum of the singular values of $I_R(G)$. We establish lower and upper bounds for the Randic incidence energy. Graphs for which these bounds are best possible are characterized. Moreover, we investigate the relation between the Randic incidence energy of a graph and that of its subgraphs. Also we give a sharp upper bound for the Randic incidence energy of a bipartite graph and determine the trees with the maximum Randic incidence energy among all $n$-vertex trees. As a result, some results are very different from those for incidence energy.


Main Subjects

1] J. A. Bondy and U. S. R. Murty, Graph Theory, 244, Springer, New York, 2008.

[2] S . B. Bozkurt and I. Gutman, Estimating the incidence energy, MATCH Commun. Math. Comput. Chem., 70 (2013) 143–156.

[3] G. Caporossi, I. Gutman, P. Hansen and L. Pavlovic, Graphs with maximum connectivity index, Comput. Biol. Chem., 27 (2003) 85–90.

[4] F. R. K. Chung, Spectral Graph Theory, Amer. Math. Soc., Providence, 1997.

[5] D. Cvetkovic, M. Doob and H. Sachs, Spectra of Graphs-Theory and Application, Third ed., Johann Ambrosius Barth. Heidelberg, 1995.

[6] D. Cvetkovic, P. Rowlinson and S. Simic, Signless Laplacians of finite graphs, Linear Algebra Appl., 423 (2007) 155–171.

[7] K. C. Das and I. Gutman, On incidence energy of graphs, Linear Algebra Appl., 446 (2014) 329–344.

[8] R. Gu, F. Huang and X. Li, General Randi´ c matrix and general Randic energy, Trans. Comb., 3 no. 3 (2014) 21-33.

[9] I. Gutman, The energy of a graph, Ber. Math.-Statist. Sekt. Forsch. Graz, 103 (1978) 1–22.

[10] I. Gutman, B. Furtula and S . B. Bozkurt, On Randic energy, Linear Algebra Appl., 442 (2014) 50–57.

[11] I. Gutman, D. Kiani and M. Mirzakhah, On incidence energy of graphs, MATCH Commun. Math. Comput. Chem., 62 (2009) 573–580.

[12] I. Gutman, D. Kiani, M. Mirzakhah and B. Zhou, On incidence energy of a graph, Linear Algebra Appl., 431 (2009) 1223–1233.

[13] I. Gutman and O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986.

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

[15] X. Li, Y. Shi and I. Gutman, Graph Energy, Springer, New York, 2012.

[16] V. Nikiforov, The energy of graphs and matrices, J. Math. Anal. Appl., 326 (2007) 1472–1475.

[17] M. Randic, On characterization of molecular branching, J. Amer. Chem. Soc., 97 (1975) 6609–6615.

[18] P. Yu, An upper bound on the Randi´ c index of trees, (Chinese), J. Math. Study, 31 (1998) 225–230.