#### Document Type

Journal Article

#### Department/Unit

Department of Mathematics

#### Title

General Randić matrix and general Randić incidence matrix

#### Abstract

© 2015 Elsevier B.V. All rights reserved. Let G be a connected graph with vertex set V(G) = {v1,⋯, vn} and edge set E(G) = {e1,⋯, em}. Let di be the degree of the vertex νi. The general Randić matrix Rα = ((Rα)ij)n×n of G is defined by (Rα)ij = (didj)^{α} if vertices νi and νj are adjacent in G and 0 otherwise. The Randić signless Laplacian matrix Qα = D^{2α+1} + Rα, where α is a nonzero real number and D is the degree diagonal matrix of G. The general Randić energy REα is the sum of absolute values of the eigenvalues of Rα. The general Randić incidence matrix BRα = ((BRα)ij)n×m of a graph G is defined by (BRα)ij = d^{α}i if νi is incident to ej and 0 otherwise. Naturally, the general Randić incidence energy BEα is the sum of the singular values of BRα. In this paper, we investigate the connected graphs with s distinct Rα-eigenvalues, where 2 ≤ s ≤ n. Moreover, we establish the relation between the Randić signless Laplacian eigenvalues of G and the general Randić energy of its subdivided graph S (G). Also we give lower and upper bounds on the general Randić incidence energy. Finally, the general Randić incidence energy of a graph and that of its subgraphs are compared.

#### Keywords

General Randić energy, General Randić incidence energy, General Randić incidence matrix, General Randić matrix

#### Source Publication Title

Discrete Applied Mathematics

#### Recommended Citation

Liu, Ruifang,
and
Wai Chee Shiu.
"General Randić matrix and general Randić incidence matrix."
*Discrete Applied Mathematics*
186
(2015): 168-175.