Alternative proof of upper bound of spanning trees in a graph

TL;DR

This paper presents a linear algebra-based proof for the upper bound on the number of spanning trees in graphs, extends it to multigraphs, and provides estimates for specific graph types, demonstrating the bound's tightness for complete graphs.

Contribution

It introduces a new proof method for the spanning tree bound, generalizes it to multigraphs, and offers estimates for specialized graph classes.

Findings

The bound is tight for complete graphs.

The proof uses linear algebra techniques.

Estimates are provided for product and Cartesian product graphs.

Abstract

We give a proof for sharp estimate for the number of spanning trees using linear algebra and generalize this bound to multigraphs. In addition, we show that this bound is tight for complete graphs. In addition, we give estimates for number of spanning trees in specific type of graphs such as product of graphs or Cartesian product of two graphs.