The problem of finding bases for the cycle-subspace and the cut-subspace is of great practical and theoretical importance. It was originally solved by Kirchhoff (1847) in his studies of electrical networks, and we shall give a brief exposition of that topic at the end of the chapter.

We shall restrict our attention to connected graphs, because the cyclesubspace and the cut-subspace of a disconnected graph are the direct sums of the corresponding spaces for the components. Throughout this chapter, Г will denote a connected graph with n vertices and m edges, so that r(Г) = n − 1 and s(Г) = m – n + 1. We shall also assume that Г has been given an orientation.

A spanning tree in Г is a subgraph which has n − 1 edges and contains no cycles. It follows that a spanning tree is connected. We shall use the symbol T to denote both the spanning tree itself and its edge-set. The following simple lemma is a direct consequence of the definition.

Lemma 5.1Let T be a spanning tree in a connected graph Г. Then:

(1) for each edge g of Г which is not in T there is a unique cycle in Г containing g and edges in T only.

(2) for each edge h of Г which is in T, there is a unique cut in Г containing h and edges not in T only.