[Paper Review] Maximum Wiener index of unicyclic graphs with given bipartition

Jan Bok, Nikola Jedličková+1 more|arXiv (Cornell University)|Feb 27, 2019

Graph theory and applications13 references4 citations

TL;DR

This paper resolves an open problem in extremal graph theory by determining the unicyclic bipartite graphs with a given bipartition (p, q) that maximize the Wiener index. It proves that the maximum is achieved uniquely by an 'onion graph' structure, specifically $\mathrm{On}(\lfloor(q-p)/2\rfloor, 2p-3, \lceil(q-p)/2\rceil)$, and provides a closed-form formula for the maximum Wiener index value.

ABSTRACT

The \emph{Wiener index} is a widely studied topological index of graphs. One of the main problems in the area is to determine which graphs of given properties attain the extremal values of Wiener index. In this paper we resolve an open problem posed by Du in 2012. To this end we determine the unicyclic bipartite graphs with given size of parts having the maximum Wiener index. This completes the previous research in which the minimum case was solved.

Motivation & Objective

• To resolve an open problem posed by Du (2012) on the maximum Wiener index for unicyclic bipartite graphs with a fixed bipartition (p, q).
• To characterize the structure of unicyclic graphs with given part sizes p and q that attain the maximum Wiener index.
• To complete the extremal characterization by solving the maximum case, following prior work on the minimum Wiener index for the same class.
• To provide a closed-form expression for the maximum Wiener index value in terms of p and q.

Proposed method

• Introduces the 'onion graph' $\mathrm{On}(k,l,m)$, defined as a 4-cycle with pendant edges on one vertex, a path of length l attached to the opposite vertex, and m pendant edges on the other end.
• Uses transmission-based analysis: the Wiener index is computed as half the sum of vertex transmissions, where transmission $t_G(v)$ is the sum of distances from v to all other vertices.
• Employs a series of lemmata to constrain the structure of extremal graphs: all extremal graphs must contain a 4-cycle and have all pendant vertices in the larger part Q.
• Applies a transformation argument using graph identification: compares Wiener indices of graphs formed by identifying vertices of smaller graphs to show that deviations from the optimal structure reduce the index.
• Uses Lemma 11 to compute exact transmissions for vertices in onion graphs, enabling precise comparison of Wiener indices.
• Applies Lemma 2 (a transmission comparison lemma) to show that any deviation from the optimal parameterization leads to a strictly larger Wiener index in a modified graph, contradicting maximality.

Experimental results

Research questions

• RQ1 Which unicyclic bipartite graphs with a given (p,q)-partition achieve the maximum Wiener index?
• RQ2 Is there a unique extremal graph structure for the maximum Wiener index in this class?
• RQ3 What is the closed-form expression for the maximum Wiener index in terms of p and q?
• RQ4 How do structural parameters like cycle size, path length, and pendant edge distribution affect the Wiener index in unicyclic graphs?

Key findings

• The maximum Wiener index for unicyclic bipartite graphs with part sizes p and q (1 < p ≤ q) is uniquely attained by the onion graph $\mathrm{On}(\lfloor(q-p)/2\rfloor, 2p-3, \lceil(q-p)/2\rceil)$.
• The Wiener index of this extremal graph is given by the explicit formula: $(2p-5)\cdot\lceil(q-p)/2\rceil\lfloor(q-p)/2\rfloor + (p-7)\lceil(q-p)/2\rceil + (13-7p)\lfloor(q-p)/2\rfloor + 2p^2q + (q-p)^2 + 2p^3 - 37p + 66$.
• Any deviation from the optimal parameterization of the onion graph (e.g., unequal distribution of pendant edges between the two cycle vertices) results in a strictly smaller Wiener index.
• The extremal graph must contain a 4-cycle, and all pendant vertices must be attached to the same cycle vertex in the larger part Q.
• The path length in the extremal graph is fixed at $2p - 3$, which is derived from the total number of vertices and the pendant edge counts.
• The proof establishes uniqueness by showing that alternative configurations either violate the bipartition constraint or yield a lower Wiener index via transmission comparisons.