Triameter of Graphs

TL;DR

This paper introduces the triameter, a new graph distance parameter, and explores its bounds, properties, and relationships with domination numbers, providing new insights and proofs in graph theory.

Contribution

It defines the triameter of a graph, establishes bounds, characterizes extremal graphs, and relates it to domination parameters, including a shorter proof for a known bound.

Findings

Bounds on triameter in terms of graph order and girth

Characterization of graphs attaining bounds

Shorter proof for a known domination number bound

Abstract

In this paper, we introduce and study a new distance parameter {\it triameter} of a connected graph $G$, which is defined as $max\{d(u,v)+d(v,w)+d(u,w): u,v,w \in V\}$ and is denoted by $tr(G)$. We find various upper and lower bounds on $tr(G)$ in terms of order, girth, domination parameters etc., and characterize the graphs attaining those bounds. In the process, we provide some lower bounds of (connected, total) domination numbers of a connected graph in terms of its triameter. The lower bound on total domination number was proved earlier by Henning and Yeo. We provide a shorter proof of that. Moreover, we prove Nordhaus-Gaddum type bounds on $tr(G)$ and find $tr(G)$ for some specific family of graphs.