TY - JOUR
T1 - General Randić matrix and general Randić incidence matrix
AU - Liu, Ruifang
AU - Shiu, Wai Chee
N1 - Supported by the National Natural Science Foundation of China (No. 11201432); the China Postdoctoral Science Foundation (Nos. 2011M501185 and 2012T50636); General Research Fund of Hong Kong (HKBU202413).
Publisher Copyright:
© 2015 Elsevier B.V. All rights reserved.
PY - 2015/5/11
Y1 - 2015/5/11
N2 - 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 = D2α+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 = dαi 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.
AB - 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 = D2α+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 = dαi 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.
KW - General Randić energy
KW - General Randić incidence energy
KW - General Randić incidence matrix
KW - General Randić matrix
UR - http://www.scopus.com/inward/record.url?scp=84933280139&partnerID=8YFLogxK
U2 - 10.1016/j.dam.2015.01.029
DO - 10.1016/j.dam.2015.01.029
M3 - Journal article
AN - SCOPUS:84933280139
SN - 0166-218X
VL - 186
SP - 168
EP - 175
JO - Discrete Applied Mathematics
JF - Discrete Applied Mathematics
IS - 1
ER -