[Paper Review] Mostar index of graph operations

Shehnaz Akhter, Zahid Iqbal+2 more|arXiv (Cornell University)|May 16, 2020

Graph theory and applications23 references7 citations

TL;DR

This paper derives closed-form formulas for the Mostar index of six graph operations—corona, Cartesian, join, lexicographic, Indu-Bala, and subdivision vertex-edge join—enabling efficient computation of the Mostar index for complex chemical graphs and nanostructures. The key contribution is a systematic framework linking topological descriptors to molecular graph operations, with explicit expressions validated on classes of chemical and nanomaterial graphs.

ABSTRACT

Very recently, a bond-additive topological descriptor, known as the Mostar index, has been proposed as a measure of peripherality in graphs and networks. In this article, we compute the Mostar index of corona product, Cartesian product, join, lexicographic product, Indu-Bala product and subdivision vertex-edge join of graphs and apply these results to find the Mostar index of various classes of chemical graphs and nanostructures.

Motivation & Objective

• To establish analytical expressions for the Mostar index of various graph operations to facilitate the study of molecular and nanostructure properties.
• To extend the applicability of the Mostar index—a bond-additive topological descriptor measuring peripherality—to complex graph constructions common in chemical and nanomaterial networks.
• To provide a computational framework that enables the efficient calculation of the Mostar index for large and intricate chemical graphs using known invariants of component graphs.
• To apply the derived formulas to compute the Mostar index for specific classes of chemical graphs and nanostructures, such as coronoids and carbon nanocones.
• To bridge theoretical graph invariants with practical applications in quantitative structure-activity and structure-property relationships (QSAR/QSPR).

Proposed method

• Derives the Mostar index for each graph operation by analyzing the contribution of each edge based on the number of vertices closer to one endpoint than the other.
• Uses vertex partitioning and distance-based counting to compute the absolute difference in vertex counts for each edge in the composite graph.
• Applies known results for base graphs (e.g., path, cycle, complete graphs) and combines them via structural decomposition to derive closed-form expressions.
• Employs case-by-case analysis for different edge types in composite graphs (e.g., edges within original graphs, edges between components, edges in subdivided or joined structures).
• Leverages symmetry and transitivity properties (e.g., vertex-transitive graphs have zero Mostar index) to simplify expressions.
• Validates results through algebraic manipulation and bounds, using inequalities to estimate contributions in complex join operations.

Experimental results

Research questions

• RQ1 How can the Mostar index be computed for the corona product of two graphs?
• RQ2 What is the Mostar index of the Cartesian product of two graphs, and how does it relate to the invariants of the factor graphs?
• RQ3 Can closed-form expressions be derived for the Mostar index of the join, lexicographic, Indu-Bala, and subdivision vertex-edge join operations?
• RQ4 How do the Mostar indices of component graphs combine in these operations to yield the index of the composite graph?
• RQ5 What are the implications of these formulas for computing the Mostar index of real-world chemical and nanostructured graphs?

Key findings

• The Mostar index of the corona product of graphs G₁ and G₂ is given by a formula involving the Mostar index of G₁, the number of vertices and edges in G₁ and G₂, and the degrees of vertices in G₂.
• For the Cartesian product G₁ □ G₂, the Mostar index is expressed as a function of the Mostar indices of G₁ and G₂, the number of vertices in each, and their respective edge counts.
• The Mostar index of the join G₁ + G₂ is derived as a combination of the Mostar indices of G₁ and G₂, the sizes of their vertex sets, and the number of edges between them.
• The lexicographic product G₁[G₂] yields a Mostar index that depends on the Mostar index of G₁, the number of vertices and edges in G₂, and the degrees of vertices in G₁.
• For the Indu-Bala product, the Mostar index is computed using a symmetric formula involving the Mostar indices of the component graphs and their structural parameters.
• The Mostar index of the subdivision vertex-edge join operation is derived through detailed case analysis of edge contributions, resulting in a formula involving the order, size, and degree sequences of the original graphs.