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Sparsity Matroid
A sparsity matroid is a mathematical structure that captures how densely a multigraph is populated with edges. To unpack this a little, sparsity is a measure of Dense graph, density of a graph that bounds the number of edges in any subgraph. The property of having a particular matroid as its density measure is invariant under graph isomorphisms and so it is a graph invariant. The graphs we are concerned with generalise simple directed graphs by allowing multiple same-oriented edges between pairs of vertices. Matroids are a quite general mathematical abstraction that describe the amount of indepdendence in, variously, points in geometric space and paths in a graph; when applied to characterising sparsity, matroids describe certain sets of sparse graphs. These matroids are connected to the structural rigidity of graphs and their ability to be decomposed into edge-disjoint spanning trees via the Tutte and Nash-Williams theorem. There is a family of efficient algorithms, known as pebb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Structure
In mathematics, a structure on a set (or on some sets) refers to providing or endowing it (or them) with certain additional features (e.g. an operation, relation, metric, or topology). Τhe additional features are attached or related to the set (or to the sets), so as to provide it (or them) with some additional meaning or significance. A partial list of possible structures are measures, algebraic structures ( groups, fields, etc.), topologies, metric structures ( geometries), orders, graphs, events, equivalence relations, differential structures, and categories. Sometimes, a set is endowed with more than one feature simultaneously, which allows mathematicians to study the interaction between the different structures more richly. For example, an ordering imposes a rigid form, shape, or topology on the set, and if a set has both a topology feature and a group feature, such that these two features are related in a certain way, then the structure becomes a topological ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
