Generalized bicycles Original Research Article

Let G = (V, E) be a graph with n vertices, and let M be a (left) module over a principal ideal domain D. Let Wo(M) and W1(M) denote the modules of all vertex and edge weightings over M, respectively. An incidence weighting η of G is a weighting of each incident vertex-edge pair (v,e) with an element of D. Now consider an orientation of the edges of G. Let δη denote the linear operator from Wo(M) to W1(M) given by δη(w)(e) = η(u, e)w(u) − η(v,e)w(v), where e ∈ E and u and v denote the tail and head of e, respectively. An η-cocycle is a weighting c ∈ W1(M), such that c = δ(w) for some w ∈ Wo(M). An η-bicycle is an η-cocycle that is also a cycle (ordinary bicycles are obtained when η ≡ 1). Let Bη(M) denote the module of all vertex weightings b ∈ Wo(M), such that δ/gh(b) is an η-bicycle. For T a spanning tree rooted at vertex v, let η(T) denote the product of η(u, e) over all the edges e of T, where u denotes the end vertex of e that is further from v in T. The η-complexity τη of G is the vertex weighting over D, such that τη(v) is the sum of η(T) over all spanning trees T rooted at v. In this paper, we show that Bη(M) ≅ M if, and only if τηm is nonzero for every nonzero m ∈ M. Further if Bη(M) ≅ M then Bη(M) = {τηm: m ∈ M}. As special cases of this formula, we obtain results about bicycles, balanced vertex weightings, resistive and leaky electrical networks, handicap ranking of tournaments, and Markov chains. The gcd η-complexity tη is the greatest common divisor of {τη(v): v ∈ V}. We show that tη has a factorization tη = t1t2 … tn − 1, unique up to multiplication by units of D, such that ti + 1 divides ti and such that for every module M over D, Bη(M) is isomorphic to the direct sum of M and the submodules M(ti) = {m ∈ M : tim = 0}, i = 1, …, n − 1. The latter formula generalizes the bicycle isomorphism formula given in Berman (1986).