Maximum number of spanning trees and connectivity: Graphs with a fixed minimum degree and bipartite graphs

TL;DR

This paper characterizes the graphs with fixed connectivity and minimum degree that maximize the number of spanning trees, including bipartite graphs, providing a comprehensive understanding of extremal structures in graph theory.

Contribution

It uniquely identifies the extremal graphs with maximum spanning trees for given connectivity, degree, and order, including bipartite cases.

Findings

Identifies graphs with maximum spanning trees for fixed connectivity and degree.

Characterizes bipartite graphs with maximum spanning trees under similar constraints.

Provides explicit extremal graph structures for these conditions.

Abstract

The number of spanning trees in a graph $G$ is the total number of distinct spanning subgraphs of $G$ that are trees. In this paper we characterize the unique graph with a prescribed vertex (resp. edge) connectivity, minimum degree and order that attains the maximum number of spanning trees. Moreover, all the bipartite graphs are determined with a given vertex (resp. edge) connectivity and order maximizing the number of spanning trees.