A relation between \(D\)-index and Wiener index for \(r\)-regular graphs (Q1989041)

Summary: For any two distinct vertices \(u\) and \(v\) in a connected graph \(G\), let \(l_P (u,v) =l (P)\) be the length of \(u-v\) path \(P\) and the \(D\)-distance between \(u\) and \(v\) of \(G\) is defined as: \(d^D (u,v)= \min_p \{l (P) + \sum_{\forall y \in V (P)} \deg y\}\), where the minimum is taken over all \(u-v\) paths \(P\) and the sum is taken over all vertices of \(u-v\) path \(P\). The \(D\)-index of \(G\) is defined as \(W^D(G)=(1/2) \sum_{\forall v,u\in V(G)} d^D (u, v)\). In this paper, we found a general formula that links the Wiener index with \(D\)-index of a regular graph \(G\). Moreover, we obtained different formulas of many special irregular graphs.