Graph construction with the maximum number of trees by continuous edge addition

Abstract

A simple computational algorithm is presented to construct a graph with the maximum number of trees by adding edges one by one. The number of trees of a graph would become an index to estimate the overall reliability of probabilistic communication networks with the same link probabilities. Our procedure, Max-trees, selects one edge that gives the maximum number of trees among edges not included in the original graph. This process is continuously repeated at each step of adding an edge, when we get the sequence of new edges to be added. As examples of the execution results, the edge sequence and the maximum number of trees are shown for two types of starting graph, which are a tree of series edges and a star-shaped tree for nodes n = 7 and 8. To see how many trees these graphs have, the minimum numbers of trees for graphs with the same number of nodes and edges are similarly calculated by the minimum-version algorithm Min-trees. An edge sequence of Max-trees makes long cycles, and that of Min-trees makes cycles of three for as long as possible. The ratio of the maximum number of trees to the minimum number of trees is about 1 to 6 for these examples.