Combinatorial Identities Using the Matrix Tree Theorem
Nayana Shibu Deepthi, Chanchal Kumar
TL;DR
This work uses (modified) matrix-tree theorem techniques to enumerate uprooted spanning trees in $K_n$ and $K_{m,n}$, yielding explicit distribution formulas and new combinatorial identities. By deriving determinant expressions from carefully constructed subgraphs $G_{n-k}$, $G_{m+n}^k$, and variants of $K_n$ with edges removed, the authors provide direct, determinant-based proofs of known counts such as $|\mathcal U_{K_n}|=(n-1)^{n-1}$ and establish new identities like $m^{n-1}n^{m-1}=\sum_{k=1}^{m} m^{n-2}(m+n-k)(n-1)^{k-1}n^{m-k-1}$. The paper further refines these enumerations by highest-child classifications and presents a refined double-sum expression for $(n-1)^{n-1}$, along with a direct proof for $|\mathcal U_{K_n}'|=(n-1)^{n-3}(n-2)^2$. The results advance the combinatorial interpretation of uprooted spanning trees and suggest extensions to more general graph classes. All results are expressed via Laplacian determinants and exact algebraic manipulations, highlighting the matrix-tree theorem as a versatile counting tool in graph enumeration.
Abstract
In this paper, we explore some interesting applications of the matrix tree theorem. In particular, we present a combinatorial interpretation of a distribution of $(n-1)^{n-1}$, in the context of uprooted spanning trees of the complete graph $K_{n}$, which was previously obtained by Chauve--Dulucq--Guibert. Additionally, we establish a combinatorial explanation for the distribution of $m^{n-1}n^{m-1}$, related to spanning trees of the complete bipartite graph $K_{m,n}$, which seems new. Furthermore, we extend this study to the graph $K_{n}\setminus \{e_{1,n}\}$, obtained by deleting an edge from $K_n$, and derive a new identity for the number of its uprooted spanning trees.