Double Hexagonal Chains with the Extremal Third Order Randic Index

Abstract

The third-order Randić index of a graph \(G\) is defined as \(R_s(G) = \sum_{u_1u_2u_3u_4} \frac{1}{\sqrt{d(u_1) d(u_2) d(u_3) d(u_4)}}\), where the summation is taken over all possible paths of length three in \(G\). In this paper, we first derive a recursive formula for computing the third-order Randić index of a double hexagonal chain. Furthermore, we establish upper and lower bounds for the third-order Randić index and characterize the double hexagonal chains that achieve the extremal third-order Randić index.