Computation of general Randi\'{c} polynomial and general Randic energy of some graphs

Abstract

The general Randi\'{c} matrix of a graph $G$, denoted by $GR(G)$ is an $n \times n$ matrix whose $(i, j)$-th entry is $(d_i d_j)^\alpha$, $\alpha \in \Bbb{R}$ if the vertices $v_i$ and $v_j$ are adjacent and $0$ otherwise, where $d_i$ is the degree of a vertex $v_i$ and $n$ is the order of $G$. 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} polynomial and the general Randi\'{c} energy of path, cycle, complete graph, complete bipartite graph, friendship graph and Dutch windmill graph.

Keywords

• general randi\'{c} eigenvalues
• general randi\'{c} energy
• randi\'{c} index
• degree of a vertex