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Gain Graph
A gain graph is a graph whose edges are labelled "invertibly", or "orientably", by elements of a group ''G''. This means that, if an edge ''e'' in one direction has label ''g'' (a group element), then in the other direction it has label ''g'' −1. The label function ''φ'' therefore has the property that it is defined differently, but not independently, on the two different orientations, or directions, of an edge ''e''. The group ''G'' is called the gain group, ''φ'' is the gain function, and the value ''φ''(''e'') is the gain of ''e'' (in some indicated direction). A gain graph is a generalization of a signed graph, where the gain group ''G'' has only two elements. See Zaslavsky (1989, 1991). A gain should not be confused with a weight on an edge, whose value is independent of the orientation of the edge. Applications Some reasons to be interested in gain graphs are their connections to network flow theory in combinatorial optimization, to geometry, and to physic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph (discrete Mathematics)
In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a Set (mathematics), set of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' means that ''A'' owes money to ''B'', then this graph is directed, because owing mon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
