Extremal results on stepwise transmission irregular graphs

TL;DR

This paper investigates extremal properties of stepwise transmission irregular graphs, identifying key families like balanced bipartite graphs and odd hatted cycles that optimize certain metrics.

Contribution

It provides new extremal results for STI graphs concerning size and metric properties, highlighting two main extremal families.

Findings

Balanced complete bipartite graphs of odd order are extremal.

Odd hatted cycles are extremal in certain cases.

Extremal properties depend on graph size and structure.

Abstract

The transmission ${\rm Tr}_G(v)$ of a vertex $v$ of a connected graph $G$ is the sum of distances between $v$ and all other vertices in $G$. $G$ is a stepwise transmission irregular (STI) graph if $|{\rm Tr}_G(u) - {\rm Tr}_G(v)| =1$ holds for each edge $uv \in E(G)$. In this paper, extremal results on STI graphs with respect to the size and different metric properties are proved. Two extremal families appear in all the cases, balanced complete bipartite graphs of odd order and the so called odd hatted cycles.