The notions of diameter, radius and center for a graph are not only of theoretical interest, but also arise naturally in applications. The lack of general methods for estimating the metric characteristics of a graph implies the necessity to create special methods for particular classes of graphs that are as a rule inapplicable outside the said class. Previously the authors have introduced a metric family of graphs, dubbed the α_i-metric. Many metric characteristics and properties of graphs were shown to be described in terms of the α_i-metric This allows to reduce the problem of computing them to the problem of finding the minimal value of i, for which the graph possesses an α_i-metric. The aim of the present work is to demonstrate the fruitfulness of this approach. We present the basic definitions and the characteristics of graphs with an α₀ and α_i-metrics. In particular we show that Ptolemaic graphs have an α₀ -metric and the chordal graphs have an α_i-metric. We also prove that finite graphs with an α_i-metric satisfy the estimate d(G)≥2r(G)-i-1 while the distance hereditary graphs and some important subclasses of the chordal graph class also satisfy the estimate d(G)≥2r(G)-i. We also investigate some properties of graph centers for graphs with an α_i-metric having i≤2.

graph theory, metric characteristics of a graph, eccentricity, α_i-metric

Mathematical problems and theory of numerical methods