# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2019: 0.755

SCImago Journal Rank (SJR) 2019: 0.600

Rejection Rate (2018-2019): c. 84%

# Discussiones Mathematicae Graph Theory

Article in press

Authors:

A. Das

Title:

Triameter of graphs

Source:

Discussiones Mathematicae Graph Theory

Received: 2018-06-05, Revised: 2019-02-25, Accepted: 2019-02-25, https://doi.org/10.7151/dmgt.2212

Abstract:

In this paper, we study a new distance parameter 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.

Keywords:

distance, radio $k$-coloring, Nordhaus-Gaddum bounds