Enumeration of spanning trees of middle graphs

TL;DR

This paper derives a formula relating the number of spanning trees in the middle graph of a digraph to the original digraph's edge weights, extending previous results on line digraphs.

Contribution

It introduces a new relation between the spanning tree counts of the middle graph and the original digraph, including a general enumerative formula for unweighted cases.

Findings

Established a relation between middle graph and original digraph complexities.

Derived an explicit enumeration formula for spanning trees in the middle graph.

Extended known results from line digraphs to middle graphs.

Abstract

Let $D$ be a connected weighted digraph. The relation between the vertex weighted complexity (with a fixed root) of the line digraph of $D$ and the edge weighted complexity (with a fixed root) of $D$ has been given in (L. Levine, Sandpile groups and spanning trees of directed line graphs, J. Combin. Theory Ser. A 118 (2011) 350-364) and, independently, in (S. Sato, New proofs for Levine's theorems, Linear Algebra Appl. 435 (2011) 943-952). In this paper, we obtain a relation between the vertex weighted complexity of the middle digraph of $D$ and the edge weighted complexity of $D$. Particularly, when the weight of each arc and each vertex of $D$ is 1, the enumerative formula of spanning trees of the middle digraph of a general digraph is obtained.