Extremal Bounds on Peripherality Measures

TL;DR

This paper establishes improved bounds on various peripherality measures in networks, including bipartite graphs and trees, and refutes two conjectures related to the Trinajstić index.

Contribution

It provides new asymptotic bounds for edge and vertex peripherality measures and disproves existing conjectures about the Trinajstić index in different graph classes.

Findings

Improved asymptotic bounds on edge and vertex peripherality measures.

Refutation of two conjectures regarding the Trinajstić index.

Results applicable to bipartite graphs, trees, and graphs with fixed diameter.

Abstract

We investigate several measures of peripherality for vertices and edges in networks. We improve asymptotic bounds on the maximum value achieved by edge peripherality, edge sum peripherality, and the Trinajsti\'c index over $n$ vertex graphs. We also prove similar results on the maxima over $n$-vertex bipartite graphs, trees, and graphs with a fixed diameter. Finally, we refute two conjectures of Furtula, the first on necessary conditions for minimizing the Trinajsti\'c index and the second about maximizing the Trinajsti\'c index.