Wiener index and Steiner 3-Wiener index of a graph

TL;DR

This paper explores the Steiner 3-Wiener index in graphs, especially modular graphs, and derives formulas relating it to the Wiener index, including specific results for Fibonacci and Lucas cubes.

Contribution

It introduces formulas connecting the Steiner 3-Wiener index with the Wiener index for modular graphs and provides explicit formulas for Fibonacci and Lucas cubes.

Findings

Steiner 3-Wiener index expressed in terms of Wiener index for modular graphs

Derived formulas for Fibonacci and Lucas cubes

Enhanced understanding of Steiner indices in specific graph classes

Abstract

Let $S$ be a set of vertices of a connected graph $G$. The Steiner distance of $S$ is the minimum size of a connected subgraph of $G$ containing all the vertices of $S$. The sum of all Steiner distances on sets of size $k$ is called the Steiner $k$-Wiener index, hence for $k=2$ we get the Wiener index. The modular graphs are graphs in which every three vertices $x, y$ and $z$ have at least one median vertex $m(x,y,z)$ that belongs to shortest paths between each pair of $x, y$ and $z$. The Steiner 3-Wiener index of a modular graph is expressed in terms of its Wiener index. As a corollary formulae for the Steiner 3-Wiener index of Fibonacci and Lucas cubes are obtained.