Abstract

For a graph $G$ of order $n$, the general Randi\'{c} matrix $GR(G)=[g_{ij}]$ is a symmetric matrix of order $n$ in which $g_{ij} = (d_id_j)^{\alpha}$, $\alpha \in \mathbb{R}$ if the vertices $v_i$ and $v_j$ are adjacent in $G$ and 0, otherwise, where $d_i$ is the degree of vertex $v_i$. The general Randi\'c energy $E_{GR}(G)$ of $G$ is the sum of the absolute values of the eigenvalues of $GR(G)$. In this paper, we compute the general Randi\'c energy of the line graph of regular graph and the graph obtained by duplication of graph elements for regular graph. We also investigate general Randi\'c equienergetic graphs.

Keywords and Phrases

General Randi\'c matrix, General Randi\'c eigenvalues, General Randi\'c energy, General Randi\'c equienergetic graphs.

A.M.S. subject classification

05C50, 05C76.

.....