The average number of spanning trees in sparse graphs with given degrees

TL;DR

This paper derives an asymptotic formula for the expected number of spanning trees in sparse graphs with specified degree sequences, under certain edge count conditions, using concentration results and Prüfer codes.

Contribution

It provides a new asymptotic expression for the expected number of spanning trees in graphs with given degrees, extending understanding of sparse graph structures.

Findings

Asymptotic expression for expected spanning trees in sparse graphs

Concentration results for functions over random trees with fixed degrees

Application of Prüfer codes in probabilistic graph analysis

Abstract

We give an asymptotic expression for the expected number of spanning trees in a random graph with a given degree sequence $\boldsymbol{d}=(d_1,\ldots, d_n)$, provided that the number of edges is at least $n + \textstyle{\frac{1}{2}} d_{\max}^4$, where $d_{\max}$ is the maximum degree. A key part of our argument involves establishing a concentration result for a certain family of functions over random trees with given degrees, using Pr\"ufer codes.