On the transmission-based graph topological indices

TL;DR

This paper introduces transmission-based topological indices for graphs, providing bounds, characterizations, and methods to compute these indices efficiently using automorphism groups.

Contribution

It defines new transmission-based indices, derives bounds, characterizes extremal graphs, and presents an automorphism-based method for efficient computation.

Findings

Derived bounds for transmission-based indices

Characterized graphs achieving bounds

Proposed automorphism group method for calculations

Abstract

The distance $d(u,v)$ between the vertices $u$ and $v$ of a connected graph $G$ is defined as the number of edges in a minimal path connecting them. The \emph{transmission} of a vertex $v$ of $G$ is defined by $\sigma(v)=\sum\limits_{u\in V(G)}{d(v,u)}$. In this article we aim to define some transmission-based topological indices. We obtain lower and upper bounds on these indices and characterize graphs for which these bounds are best possible. Finally, we find these indices for various graphs using the group of automorphisms of $G$. This is an efficient method of finding these indices especially when the automorphism group of $G$ has a few orbits on $V(G)$ or $E(G)$.