Express the number of spanning trees in term of degrees

TL;DR

This paper introduces a new method to compute the number of spanning trees in a connected graph using vertex degrees, offering an alternative to existing theorems like Matrix-Tree and deletion-contraction.

Contribution

It presents an alternative approach to calculating spanning trees based on vertex degrees, expanding the computational tools available.

Findings

Provides a new degree-based formula for spanning trees

Offers an alternative to classical theorems

Simplifies computation in certain cases

Abstract

It is well-known that the number of spanning trees, denoted by $\tau(G)$, in a connected multi-graph $G$ can be calculated by the Matrix-Tree theorem and Tutte's deletion-contraction theorem. In this short note, we find an alternate method to compute $\tau(G)$ by degrees of vertices.