On Clique Version of the Randic Index

TL;DR

This paper introduces a generalized clique-based Randic index, extending classical graph invariants, and establishes an upper bound for this new measure, opening avenues for further research in graph theory.

Contribution

It presents a novel clique-based generalization of the Randic index and proves an upper bound for this new measure, expanding theoretical understanding.

Findings

Established a generalized version of the Randic index.

Proved an upper bound for the generalized Randic index.

Discussed potential directions for future research.

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.