The Cycle Spaces of an Infinite Graph

Abstract

The edge space of a finite graph <em>G</em> = (<em>V</em>, <em>E</em>) over a field F is simply an assignment of field elements to the edges of the graph. The edge space can equally be thought of us an |<em>E</em>|-dimensional vector space over F. The cycle space and bond space are the subspaces of the edge space generated by the cycle and bonds of the graph respectively. It is easy to prove that the cycle space and bond space are orthogonal complements. <br /><br /> Unfortunately many of the basic results in finite dimensional vector spaces no longer hold in infinite dimensions. Therefore extending the cycle and bond spaces to infinite graphs is not at all a trivial exercise. <br /><br /> 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 allows us to enrich the set of cycles to include infinite cycles. We introduce two cycle spaces and three bond spaces of a locally finite graph and determine the orthogonality relations between them. We also determine the sum of two of these spaces, and derive a version of the Edge Tripartition Theorem.