Typical metric properties of n-vertex graphs of given diameter

Abstract

When studying a given class of graphs that admit the concept of dimension, i.e., a measure of their number (often, the dimension of a graph is understood as the number of its vertices; of course, there are other approaches), questions of an asymptotic nature naturally arise. In the asymptotic study of the class Ω_n of graphs of dimension n, special attention is drawn to the topics around the following three questions. The first one is computing the asymptotically exact value of the number of such graphs (or obtaining good estimates for it). This makes it quite easy to calculate, as a rule, the hard-to-calculate number Ω_n with a specified accuracy. The second issue is isolating or subclassing generic graphs Ω_n^* ⊆ Ω_n for the given Ω_n class. And the third is the study of general, typical properties (valid for almost all) of the graphs under consideration. This approach significantly helps to understand the structure of graphs of the entire class, especially with a large number of vertices. The report discusses the designated topic for the class of n-vertex graphs of a given diameter. We study the typical metric properties of these graphs related to the variety of metric balls, the graph radius, diametrical and central vertices, the graph center and its spectrum (the cardinality set of graph centers), etc., as well as the classes of typical graphs that arise here.