Bicycles and Spanning Trees

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 ∈ 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 = t₁ t₂ ⋯ tₘ such that tᵢ is a multiple of ti + 1 ,i = 1,2, ⋯ ,m - 1 and such that for every Abelian group A the group $B ( A )$ of bicycles over A is isomorphic to A ( t₁ ) × A ( t₂ ) × ⋯ × A ( tₘ ). Using this result we obtain a number of results on the spanning tree number including two formulae for the spanning tree number.