Proof of a Conjecture on the Wiener Index of Eulerian Graphs

TL;DR

This paper proves a conjecture stating that for large Eulerian graphs, the structure with the second largest Wiener index is a cycle with a shared vertex triangle, confirming previous partial results.

Contribution

The paper proves the conjecture that a specific graph structure maximizes the Wiener index among Eulerian graphs of large order.

Findings

Cycle graph uniquely maximizes Wiener index among Eulerian graphs.

The second largest Wiener index is achieved by a cycle with a shared vertex triangle for large n.

Conjecture holds for all n ≥ 26, completing previous partial proofs.

Abstract

The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices. A connected graph is Eulerian if its vertex degrees are all even. In [Gutman, Cruz, Rada, Wiener index of Eulerian Graphs, Discrete Applied Mathematics 132 (2014), 247-250] the authors proved that the cycle is the unique graph maximising the Wiener index among all Eulerian graphs of given order. They also conjectured that for Eulerian graphs of order $n \geq 26$ the graph consisting of a cycle on $n-2$ vertices and a triangle that share a vertex is the unique Eulerian graph with second largest Wiener index. The conjecture is known to hold for all $n\leq 25$ with exception of six values. In this paper we prove the conjecture.