[Paper Review] Resolving Share and Topological Index

Muhammad Salman, Imran Javaid+1 more|arXiv (Cornell University)|Aug 1, 2014

Graph theory and applications19 references3 citations

TL;DR

This paper introduces the resolving topological index, a novel distance-based topological index derived from the resolving share of each vertex in a molecular graph. By quantifying how much each atom contributes to uniquely distinguishing pairs of atoms via distance differences, the index captures the graph's topology through resolvability behavior, with explicit computations provided for paths, cycles, wheels, complete graphs, complete k-partite graphs, and friendship graphs.

ABSTRACT

An atom $a$ of a molecular graph $G$ uniquely determines (resolves) a pair $(a_1,a_2)$ of atoms of $G$ if the distance between $a$ and $a_1$ is different from the distance between $a$ and $a_2$. In this paper, we quantify the involvement of each atom $a$ of $G$ in uniquely determining (resolving) a pair $(a_1,a_2)$ of atoms of $G$, which is called the resolving share of $a$ for the pair $(a_1,a_2)$. Using this quantity, we define a distance-based topological index of a molecular graph, which reflects the topology of that molecular graph according to the resolvability behavior of each of its atom, and is called the resolving topological index. Then we compute the resolving topological index of several molecular graphs.

Motivation & Objective

• To quantify the contribution of each vertex in a molecular graph to uniquely resolving all pairs of vertices via distance differences.
• To define a new topological index—called the resolving topological index—that reflects the overall resolvability behavior of a molecular graph.
• To compute the resolving topological index for several well-known molecular graphs, including paths, cycles, wheels, friendship graphs, and complete k-partite graphs.
• To introduce and analyze the average resolving share of a vertex as a measure of its relative resolving influence across all pairs of vertices.

Proposed method

• Define the resolving share $ r_w(u,v) $ of a vertex $ w $ for a pair $ (u,v) $ as $ \frac{1}{2} $ if $ w $ resolves the pair and $ 0 $ otherwise, with normalization to ensure sum over all vertices equals 1.
• Use the resolving share to compute the average resolving share $ ar_v(G) $ for each vertex $ v $, representing its normalized contribution to resolving all pairs.
• Construct the resolving topological index $ \mathcal{R}(G) $ as the sum of the average resolving shares over all vertices in $ G $, i.e., $ \mathcal{R}(G) = \sum_{v \in V(G)} ar_v(G) $.
• Apply the framework to compute $ \mathcal{R}(G) $ for specific graphs using structural properties such as diameter, vertex partitioning, and symmetry.
• Leverage lemmas on resolving sets and distance partitions to derive closed-form expressions for resolving shares in symmetric graphs like wheels and friendship graphs.
• Use the join operation $ G_1 + G_2 $ to model complex graphs such as friendship graphs and derive resolving shares based on vertex partitioning and distance behavior.

Experimental results

Research questions

• RQ1 How much does each vertex in a molecular graph contribute to resolving all pairs of vertices via distance differences?
• RQ2 What is the total resolving capacity of a molecular graph, measured as the sum of normalized resolving contributions of all its vertices?
• RQ3 How does the resolving topological index vary across different classes of molecular graphs, such as paths, cycles, and complete graphs?
• RQ4 Can the resolving topological index be computed in closed form for symmetric molecular graphs like wheels and friendship graphs?
• RQ5 What is the relationship between the average resolving share of a vertex and its structural position in the graph?

Key findings

• The resolving topological index of a path $ P_n $ is $ \frac{n^2 + 8}{6(n-2)(4n-13)} $ for $ n \geq 4 $, derived from the average resolving shares of its vertices.
• The resolving topological index of a cycle $ C_n $ is $ \frac{n^2 - 2n - 2}{2(n-2)(4n-13)} $, valid for $ n \geq 4 $, reflecting its symmetric resolving behavior.
• For a wheel graph $ W_n $, the resolving topological index is $ \frac{(n-3)(n^2 + 8)}{6(n-2)(4n-13)} $, computed using vertex-specific resolving shares.
• The resolving topological index of a friendship graph $ F_n $ is $ \frac{2n^3 - n^2 + 4n - 4}{4n(3n - 2)} $, with contributions from the central vertex and peripheral vertices.
• The average resolving share of the central vertex in $ F_n $ is $ \frac{1}{2n} $, while that of peripheral vertices is $ \frac{(n+1)(2n-1)}{4n(3n-2)} $, showing its dependence on graph size.
• The resolving topological index provides a new, numerically computable topological invariant that reflects the global resolvability structure of a molecular graph.