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 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.
Original language | English |
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Pages (from-to) | 168-175 |
Number of pages | 8 |
Journal | Discrete Applied Mathematics |
Volume | 186 |
Issue number | 1 |
DOIs | |
Publication status | Published - 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