A New Proof of Cayley’s Formula for Labeled Spanning Trees M. Ariannejad, M. Emami Department of Mathematics, University of Zanjan Zanjan, I. R. Iran Abstract Cayleys’s formula for enumerating spanning trees in complete graphs is one of the main theorems in this topic. In this note based on a new recursive method for the enumeration of spanning trees and meanwhile enumerating spanning trees in a kind of complete multigraph, a new proof for the Cayley’s formula is given. A generalization of this formula is also presented. Keywords: Cayleys’s formula, Spanning tree. 1 Introduction One of the classical topics in any graph theory texts is enumerating spanning trees in diverse kinds of graphs. Let G be a given graph. By τ (G) we denote the total number of spanning trees in G. We also denote the complete simple graph with n vertex by K n . Cayley’s formula [1] is one of the best computa- tions in this title: Theorem 1.1 (Cayley’s Formula[1889]) τ (K n )= n n-2 . There is also a simple recursive formula for computing the number of spanning Available online at www.sciencedirect.com Electronic Notes in Discrete Mathematics 45 (2014) 99–102 1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. www.elsevier.com/locate/endm http://dx.doi.org/10.1016/j.endm.2013.11.019