The number of spanning trees in the lexicographic product of path and any graph

In the fields of graph theory and network science, the number of spanning trees is a key indicator for evaluating the topological structure of a network. The number of spanning trees directly reflects a network's connectivity and fault tolerance in the presence of node or edge failures. Therefore, computing the number of spanning trees in a graph has practical significance for evaluating the reliability of supercomputer networks. In this paper, we focus on the problem of counting spanning trees in the lexicographic product of a path and a simple connected undirected graph. By transforming the combinatorial problem into an algebraic one, we derive a closed-form formula for the number of spanning trees. This research not only deepens the understanding of the structural properties of lexicographic product graphs but also provides a solid theoretical foundation for the design and optimization of supercomputing networks.