## Abstract

Let G be a connected graph with vertex set V(G) = {v_{1},⋯, v_{n}} and edge set E(G) = {e_{1},⋯, e_{m}}. Let d_{i} be the degree of the vertex ν_{i}. The general Randić matrix R_{α} = ((R_{α})_{ij})n×n of G is defined by (R_{α})_{ij} = (d_{i}d_{j})^{α} if vertices ν_{i} and ν_{j} are adjacent in G and 0 otherwise. The Randić signless Laplacian matrix Q_{α} = D^{2α+1} + R_{α}, where α is a nonzero real number and D is the degree diagonal matrix of G. The general Randić energy RE_{α} is the sum of absolute values of the eigenvalues of R_{α}. The general Randić incidence matrix B_{Rα} = ((B_{Rα})_{ij})n×m of a graph G is defined by (B_{Rα})_{ij} = d^{α}_{i} if ν_{i} is incident to e_{j} and 0 otherwise. Naturally, the general Randić incidence energy BE_{α} is the sum of the singular values of B_{Rα}. In this paper, we investigate the connected graphs with s distinct R_{α}-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 - 11 May 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