Counting Spanning Trees of Threshold Graphs

Abstract

Cayley's formula states that there are n n−2 spanning trees in the complete graph on n vertices; it has been proved in more than a dozen different ways over its 150 year history. The complete graphs are a special case of threshold graphs, and using Merris' Theorem and the Matrix Tree Theorem, there is a strikingly simple formula for counting the number of spanning trees in a threshold graph on n vertices; it is simply the product, over i = 2, 3, . . . , n − 1, of the number of vertices of degree at least i. In this manuscript, we provide a direct combinatorial proof for this formula which does not use the Matrix Tree Theorem; the proof is an extension of Joyal's proof for Cayley's formula. Then we apply this methodology to give a formula for the number of spanning trees in any difference graph.

Linear Algebraic Techniques for Spanning Tree Enumeration

“…As with Ferrers graphs, spanning trees in threshold can also be counted by a beautiful formula, which appears several places in the literature, including proofs by Chestnut and Fishkind [5], Hammer and Kelmans [7], and Merris [14].…”