[Paper Review] New methods for calculating the degree distance and the Gutman index

Simon Brezovnik, Niko Tratnik|arXiv (Cornell University)|Sep 7, 2018

Graph theory and applications19 references3 citations

TL;DR

This paper introduces novel computational methods for the degree distance and Gutman index using weighted quotient graphs derived from the Θ*-partition of a graph’s edge set. It establishes that these indices can be computed via Wiener indices of quotient graphs, with a key result showing equality in the Gutman index bound if and only if the graph is a partial Hamming graph, enabling exact formulas for specific families like Hₙ.

ABSTRACT

In the paper we develop new methods for calculating the two well-known topological indices, the degree-distance and the Gutman index. Firstly, we prove that the Wiener index of a double vertex-weighted graph can be computed from the Wiener indices of weighted quotient graphs with respect to a partition of the edge set that is coarser than $Θ^*$-partition. This result immediately gives a method for computing the degree-distance of any graph. Next, we express the degree-distance and the Gutman index of an arbitrary phenylene by using its hexagonal squeeze and inner dual. In addition, it is shown how these two indices of a phenylene can be obtained from the four quotient trees. Furthermore, reduction theorems for the Wiener index of a double vertex-weighted graph are presented. Finally, a formula for computing the Gutman index of a partial Hamming graph is obtained.

Motivation & Objective

• To develop generalizable methods for computing the degree distance and Gutman index of arbitrary graphs.
• To extend cut-method techniques to double vertex-weighted graphs and apply them to topological indices.
• To express the degree distance and Gutman index of phenylenes using their hexagonal squeeze and inner dual.
• To derive reduction theorems for the Wiener index of double vertex-weighted graphs to simplify index computation.
• To establish a tight inequality for the Gutman index and characterize equality conditions via partial Hamming graphs.

Proposed method

• Utilizes the Θ*-partition of a graph’s edge set to define quotient graphs, enabling decomposition of the Wiener index computation.
• Applies a generalized cut method to compute the Wiener index of double vertex-weighted graphs using quotient graphs with respect to coarser partitions than Θ*.
• Expresses the degree distance and Gutman index of phenylenes through their hexagonal squeeze and inner dual, leveraging quotient trees.
• Derives reduction theorems for the Wiener index of double vertex-weighted graphs by analyzing component-wise weight sums across Θ*-edge removal.
• Applies canonical isometric embedding to relate the Wiener index to quotient graph completeness, leading to an inequality with equality iff the graph is a partial Hamming graph.
• Uses the equality condition in the inequality to derive a closed-form formula for the Gutman index of Hₙ, an infinite family of partial Hamming graphs.

Experimental results

Research questions

• RQ1 Can the degree distance and Gutman index be computed efficiently using quotient graphs derived from the Θ*-partition?
• RQ2 How can the Wiener index of a double vertex-weighted graph be reduced using quotient structures?
• RQ3 What is the relationship between the Gutman index and the structure of partial Hamming graphs?
• RQ4 Can the degree distance and Gutman index of phenylenes be expressed in terms of their hexagonal squeeze and inner dual?
• RQ5 Under what conditions does the inequality for the Wiener index of a weighted graph become an equality?

Key findings

• The degree distance and Gutman index of any graph can be computed via the Wiener indices of its quotient graphs with respect to a partition coarser than the Θ*-partition.
• For phenylenes, both the degree distance and Gutman index are expressible using their hexagonal squeeze and inner dual, with explicit computation possible from four quotient trees.
• A reduction theorem for the Wiener index of double vertex-weighted graphs is established, enabling recursive or component-based computation.
• An inequality for the Wiener index of a weighted graph is proven, with equality if and only if the graph is a partial Hamming graph.
• The Gutman index of the infinite family Hₙ (n ≥ 2) is exactly 6n³ + 9n² − 4n + 1, derived from the equality condition in the inequality.
• The quotient graphs of Hₙ with respect to Θ*-classes are all complete, confirming Hₙ is a partial Hamming graph, validating the equality condition.