On Clique Version of the Randic Index

Abstract

In this paper, we first review the weighted-versiion of the handshaking lemma based on the idea of a weighted vertex-edge incidence matrix of a given graph $G$. Then, we obtain a generalized version of the handshaking lemma based on the concept of the clique value. We also define a generalized version of Randic index. More importantly, we prove an upper bound for the generalized Randic index of a graph $G$. We finally concluse the paper with some disscussions about possible future works.