General Randić matrix and general Randić incidence matrix

Ruifang Liu, Wai Chee SHIU*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)

Abstract

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 B = ((B)ij)n×m of a graph G is defined by (B)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 B. 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.

Original languageEnglish
Pages (from-to)168-175
Number of pages8
JournalDiscrete Applied Mathematics
Volume186
Issue number1
DOIs
Publication statusPublished - 2015

Scopus Subject Areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

User-Defined Keywords

  • General Randić energy
  • General Randić incidence energy
  • General Randić incidence matrix
  • General Randić matrix

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