13,485 research outputs found
The Cycle Spaces of an Infinite Graph
The edge space of a finite graph G = (V, E) 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 |E|-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. 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. 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
Infinite matroids in graphs
It has recently been shown that infinite matroids can be axiomatized in a way
that is very similar to finite matroids and permits duality. This was
previously thought impossible, since finitary infinite matroids must have
non-finitary duals. In this paper we illustrate the new theory by exhibiting
its implications for the cycle and bond matroids of infinite graphs. We also
describe their algebraic cycle matroids, those whose circuits are the finite
cycles and double rays, and determine their duals. Finally, we give a
sufficient condition for a matroid to be representable in a sense adapted to
infinite matroids. Which graphic matroids are representable in this sense
remains an open question.Comment: Figure correcte
Infinite graphic matroids Part I
An infinite matroid is graphic if all of its finite minors are graphic and
the intersection of any circuit with any cocircuit is finite. We show that a
matroid is graphic if and only if it can be represented by a graph-like
topological space: that is, a graph-like space in the sense of Thomassen and
Vella. This extends Tutte's characterization of finite graphic matroids.
The representation we construct has many pleasant topological properties.
Working in the representing space, we prove that any circuit in a 3-connected
graphic matroid is countable
Axioms for infinite matroids
We give axiomatic foundations for non-finitary infinite matroids with
duality, in terms of independent sets, bases, circuits, closure and rank. This
completes the solution to a problem of Rado of 1966.Comment: 33 pp., 2 fig
Dirac Operators and the Calculation of the Connes Metric on arbitrary (Infinite) Graphs
As an outgrowth of our investigation of non-regular spaces within the context
of quantum gravity and non-commutative geometry, we develop a graph Hilbert
space framework on arbitrary (infinite) graphs and use it to study spectral
properties of graph-Laplacians and graph-Dirac-operators. We define a spectral
triplet sharing most of the properties of what Connes calls a spectral triple.
With the help of this scheme we derive an explicit expression for the
Connes-distance function on general directed or undirected graphs. We derive a
series of apriori estimates and calculate it for a variety of examples of
graphs. As a possibly interesting aside, we show that the natural setting of
approaching such problems may be the framework of (non-)linear programming or
optimization. We compare our results (arrived at within our particular
framework) with the results of other authors and show that the seeming
differences depend on the use of different graph-geometries and/or Dirac
operators.Comment: 27 pages, Latex, comlementary to an earlier paper, general treatment
of directed and undirected graphs, in section 4 a series of general results
and estimates concerning the Connes Distance on graphs together with examples
and numerical estimate
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