Combinatorial Identities Using the Matrix Tree Theorem

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.