On radius and typical properties of $n$-vertex graphs of given diameter

Abstract

A property of graphs from a class under consideration is typical if almost all graphs from this class have the given property. Typical properties of the class of n-vertex graphs of a xed diameter k are studied. A family of embedded classes of typical n-vertex graphs of a given diameter k ≥ 3, which possess a number of established metric properties, is constructed. Based on the typical properties of metric balls contained in the graph, the radius of almost all n-vertex graphs from the investigated classes is found. It is proved that for every xed integer k ≥ 3 almost all n-vertex graphs of diameter k have radius d k 2 e, while the radius of almost all graphs of diameter k = 1, 2 is equal to the diameter. All found typical properties of n-vertex graphs of a xed diameter k ≥ 2 are also typical for connected graphs of diameter at least k, as well as for graphs (not necessarily connected) containing the shortest path of length at least k.