
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 , which is defined as and is denoted by . We find various upper and lower bounds on 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 and find for some specific family of graphs.
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