Peripherality in networks: theory and applications

TL;DR

This paper explores measures of peripherality and centrality in networks, focusing on the Mostar index, refuting previous conjectures, correcting results, and applying these concepts to real-world chemical reaction graphs.

Contribution

It introduces new bounds, complexity results, and indices for measuring peripherality, and evaluates the Mostar index's effectiveness in different network contexts.

Findings

Refutes a conjecture on the maximum Mostar index of bipartite graphs.

Corrects a maximum terminal Mostar index value for trees, showing it is (n-1)(n-3).

Finds Mostar index aligns more with centrality than peripherality in chemical reaction graphs.

Abstract

We investigate several related measures of peripherality and centrality for vertices and edges in networks, including the Mostar index which was recently introduced as a measure of peripherality for both edges and networks. We refute a conjecture on the maximum possible Mostar index of bipartite graphs from (Do\v{s}li\'{c} et al, Journal of Mathematical Chemistry, 2018) and (Ali and Do\v{s}li\'{c}, Applied Mathematics and Computation, 2021). We also correct a result from the latter paper, where they claimed that the maximum possible value of the terminal Mostar index among all trees of order $n$ is $(n-1)(n-2)$. We show that this maximum is $(n-1)(n-3)$ for $n \ge 3$, and that it is only attained by the star. We asymptotically answer another problem on the maximum difference between the Mostar index and the irregularity of trees from (F. Gao et al, On the difference of Mostar index and irregularity of graphs, Bulletin of the Malaysian Mathematical Sciences Society, 2021). We also prove a number of extremal bounds and computational complexity results about the Mostar index, irregularity, and measures of peripherality and centrality. We discuss graphs where the Mostar index is not an accurate measure of peripherality. We construct a general family of graphs with the property that the Mostar index is strictly greater for edges that are closer to the center. We also investigate centrality and peripherality in two graphs which represent the SuperFast and MOZART-4 systems of atmospheric chemical reactions by computing various measures of peripherality and centrality for the vertices and edges in these graphs. For both of these graphs, we find that the Mostar index is closer to a measure of centrality than peripherality of the edges. We also introduce some new indices which perform well as measures of peripherality on the SuperFast and MOZART-4 graphs.