Maximum number of spanning trees in bipartite graphs with a given diameter

Abstract

The number of spanning trees is a classical graph invariant and an important measure of network reliability, as it counts the minimal connected spanning substructures that can maintain communication in a network. Let $\mathcal{B}(n,d)$ be the set of connected bipartite graphs of order $n$ and diameter $d$. Motivated by reliability design problems for bipartite network models with fixed order and diameter, this paper determines all graphs with the maximum number of spanning trees in $\mathcal{B}(n,d)$. The result gives an extremal characterization of bipartite network topologies with the largest number of connected spanning backbones under prescribed order and diameter constraints, and provides a structural reference for the design of reliable bipartite networks.