Bicycles and Spanning Trees

Abstract

Let G be a connected multigraph and let $( A, + ,0 )$ be any Abelian group. For k an integer, let $A ( k )$ denote the subgroup of A given by $A ( k ) = \{ a \in A | ka = 0 \}$. A bicycle over A is a cycle over A that is also a cocycle. The set $B ( A )$ of bicycles over A determines a group. In this paper we show that the spanning tree number t of G has a unique factorization $t = t_1 t_2 \cdots t_m $ such that $t_i $ is a multiple of $t_{i + 1} ,i = 1,2, \cdots ,m - 1$ and such that for every Abelian group A the group $B ( A )$ of bicycles over A is isomorphic to $A ( t_1 ) \times A ( t_2 ) \times \cdots \times A ( t_m )$. Using this result we obtain a number of results on the spanning tree number including two formulae for the spanning tree number.