Two problems in the search of metric characteristics on weighted undirected graphs with non-negative edge weights are being considered. The first problem: a weighted undirected graph with non-negative edge weight is given. The radius, diameter and at least one center and one pair of peripheral vertices of the graph are to be found. In the second problem we have additionally calculated the distances matrix. For the problems being considered, we proposed fast search algorithms which use only small fraction of graph's vertices for the search of the metric characteristics. The proposed algorithms have been compared to other popular methods of solving problems considered on various inputs.
In this paper we present a novel approach to determine the exact diameter (longest shortest path length) of large graphs, in particular of the nowadays frequently studied small world networks. Typical examples include social networks, gene networks, web graphs and internet topology networks. Due to complexity issues, the diameter is often calculated based on a sample of only a fraction of the nodes in the graph, or some approximation algorithm is applied. We instead propose an exact algorithm that uses various lower and upper bounds as well as effective node selection and pruning strategies in order to evaluate only the critical nodes which ultimately determine the diameter. We will show that our algorithm is able to quickly determine the exact diameter of various large datasets of small world networks with millions of nodes and hundreds of millions of links, whereas before only approximations could be given. In this note, we present first linear time algorithms for computing the center and the diameter of several classes of face regular plane graphs: triangulations with inner vertices of degree ≥ 6, quadrangulations with inner vertices of degree ≥ 4 and the subgraphs of the regular hexagonal grid bounded by a simple circuit of this grid.