The number of spanning trees in a new lexicographic product of graphs

Abstract

The problem of counting the number of spanning trees is an old topic in graph theory with important applications to reliable network design. Usually, it is desirable to put forward a formula of the number of spanning trees for various graphs, which is not only interesting in its own right but also in practice. Since some large graphs can be composed of some existing smaller graphs by using the product of graphs, the number of spanning trees of such large graph is also closely related to that of the corresponding smaller ones. In this article, we establish a formula for the number of spanning trees in the lexicographic product of two graphs, in which one graph is an arbitrary graph G and the other is a complete multipartite graph. The results extend some of the previous work, which is closely related to the number of vertices and Lapalacian eigenvalues of smaller graphs only.

On forwarding indices of lexicographic product networks

“…More on the theory and application of the product graphs; see other works. [26][27][28][29][30][31][32][33] The lexicographic product method not only provides us with a useful technique to solve the problem of network topology design but also has its own theoretical research value, especially serving as a link between the two subjects related to the graph and the network design. There are many very famous journals dealing with this article.…”

“…Therefore, this method has become an important method for designing high performance super‐computers and large‐scale interconnection networks. More on the theory and application of the product graphs; see other works 26–33 …”

Minimum strong radius of the strong product of paths

“…In [11], Chen provided an orientation method for the minimum strong radius of the cartesian product, as shown in Figure 1. And the red vertices in the figure 1…”

“…From the definition, we can see that in the case of the same factor graph, the cartesian product graph is a subgraph of the strong product graph. At present, the research on product graphs is very rich [11][12][13][14][15][16][17].…”