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arXiv · 2504.21319

Combinatorial Identities Using the Matrix Tree Theorem

Nayana Shibu Deepthi·Chanchal Kumar
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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.

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Nayana Shibu Deepthi, Chanchal Kumar. 2025-04-30. Combinatorial Identities Using the Matrix Tree Theorem. https://arxiv.org/abs/2504.21319

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