Abstract

In many modern areas of automatic design, problems in the synthesis and analysis of stable electronic circuits arise, which can be geometrically presented by directed or undirected graphs, in which the number of the shortest paths between the given vertices plays a significant role. The article investigates the problem of finding the number of shortest paths connecting vertices s and t of the graph G. For this graph, the concepts of vertex and edge (s, t)-cuts are defined and among these cuts simple cuts are highlighted. The concepts of graphs (s, t)-critical with respect to vertices and (s, t)-critical with respect to edges are also defined. Statements are given based on the known rules in combinatorial analysis sum and product and their generalizations, by which we can find the number of the shortest (s, t)-chains of the graph G through the given vertex (s, t)-cut or the given edge (s, t)-cut. The paper considers, in particular, the graph of a square grid of the size n × m, which is (s, t)-critical with respect to both vertices and edges, where s and t are the opposite vertices of this grid. For such a grid we have described all possible vertex (s, t)-cuts, which are simple. The edge (s, t)-cuts of this grid are also described, which are determined by the shortest paths of the dual graph, which is also a square grid of size (n+1) × (m+1). From the considered cuts for the number of combinations ((n+m)¦n) equal to ((n+m)!)/n!m! we get ((n+m+2)!)/((n+1)!(m+1)!) + n + m – 3 combinatorial identities. The above statements can also be applied to study the number of shortest paths in multigraphs – graphs containing parallel edges. Based on the canonical decomposition of a natural number a into prime factors, an optimal multigraph is shown in which the number of the shortest (s, t)-paths is equal to a given number a.

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