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When modelling relations between two different classes of objects, bipartite graphs very often arise naturally. For instance, a graph of football players and clubs, with an edge between a player and a club if the player has played for that club, is a natural example of an affiliation network, a type of bipartite graph used in social network analysis.
The complete bipartite graph K m,n has a vertex covering number of min{m, n} and an edge covering number of max{m, n}. The complete bipartite graph K m,n has a maximum independent set of size max{m, n}. The adjacency matrix of a complete bipartite graph K m,n has eigenvalues √ nm, − √ nm and 0; with multiplicity 1, 1 and n + m − 2 ...
A stronger definition of bipartiteness is: a hypergraph is called bipartite if its vertex set V can be partitioned into two sets, X and Y, such that each hyperedge contains exactly one element of X. [2] [3] Every bipartite graph is also a bipartite hypergraph. Every bipartite hypergraph is 2-colorable, but bipartiteness is stronger than 2 ...
A further generalization is the f-factor problem for bipartite graphs, i.e. for a given bipartite graph one searches for a subgraph possessing a certain degree sequence. The problem uniform sampling a bipartite graph to a fixed degree sequence is to construct a solution for the bipartite realization problem with the additional constraint that ...
In the mathematical field of graph theory, a convex bipartite graph is a bipartite graph with specific properties. A bipartite graph, (U ∪ V, E), is said to be convex over the vertex set U if U can be enumerated such that for all v ∈ V the vertices adjacent to v are consecutive. Convexity over V is defined analogously. A bipartite graph (U ...
A bipartite graph with 4 vertices on each side, 13 edges, and no , subgraph, and an equivalent set of 13 points in a 4 × 4 grid, showing that (;).. The number (;) asks for the maximum number of edges in a bipartite graph with vertices on each side that has no 4-cycle (its girth is six or more).
An example is the "Möbius strip" graph ,, formed by removing a -cycle from the complete bipartite graph with parts of size . László Lovász proved a local version of Sidorenko's conjecture, i.e. for graphs that are "close" to random graphs in a sense of cut norm.
The Birkhoff polytope B n (also called the assignment polytope, the polytope of doubly stochastic matrices, or the perfect matching polytope of the complete bipartite graph , [1]) is the convex polytope in R N (where N = n 2) whose points are the doubly stochastic matrices, i.e., the n × n matrices whose entries are non-negative real numbers and whose rows and columns each add up to 1.