The Cycle Spaces of an Infinite Graph

[摘要] The edge space of a finite graph G = (V, E) over a field F is simply anassignment of field elements to the edges of the graph. The edge space canequally be thought of us an |E|-dimensional vector space over F. The cycle space andbond space are the subspaces of the edge space generated by the cycle and bondsof the graph respectively. It is easy to prove that the cycle space and bondspace are orthogonal complements.

Unfortunately many of the basic results in finite dimensional vector spaces nolonger hold in infinite dimensions. Therefore extending the cycle and bondspaces to infinite graphs is not at all a trivial exercise.

This thesis is mainly concerned with the algebraic properties of the cycle and bond spaces of a locally finite, infinite graph. Our approach is to first topologize and then compactify the graph. This allowsus to enrich the set of cycles to include infinite cycles. We introduce twocycle spaces and three bond spaces of a locally finitegraph and determine the orthogonality relations between them.We also determine the sum of two of these spaces, and derive a version of theEdge Tripartition Theorem.

[发布机构] University of Waterloo

[关键词] Mathematics