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 Levine [Sandpile groups and spanning trees of directed line graphs. J Combin Theory Ser A. 2011;118:350–364] and, independently, in Sato [New proofs for Levine's theorems. Linear Algebra Appl. 2011;435:943–952]. 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.
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