Algebraic Graph Theory Without Orientation

Let G be an undirected graph with vertices {v1, v2, . . . , vν} and edges {e1, e2, . . . , eε}. Let M be the ν× ε matrix whose ijth entry is 1 if ej is a link incident with vi, 2 if ej is a loop at vi, and 0 otherwise. The matrix obtained by orienting the edges of a loopless graph G (i.e., changing one of the 1s to a −1 in each column of M) has been studied extensively in the literature. The purpose of this paper is to explore the substructures of G and the vector spaces associated with the matrix M without imposing such an orientation. We describe explicitly bases for the kernel and range of the linear transformation from R to R defined by M. Our main results are determinantal formulas, using the unoriented Laplacian matrix MM, to count certain spanning substructures of G. These formulas may be viewed as generalizations of the Matrix Tree Theorem. The point of view adopted in this paper also gives rise to a matroid structure on the edges of G analogous to the cycle matroid and its dual. In this setting, the analogue of a spanning forest can have components with one odd cycle, and the analogue of an edge cut has the property that its removal creates a new bipartite component. AMS Subject Classification (1991): Primary: 05C50 Secondary: 05B35, 05C05, 05C30, 05C38, 15A03