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The circuit rank of a hypergraph can be derived by its Levi graph, with the same circuit rank but reduced to a simple graph. = + (+) where g is the degree sum, e is the number of edges in the given graph, v is the number of vertices, and c is the number of connected components.
This graph becomes disconnected when the right-most node in the gray area on the left is removed This graph becomes disconnected when the dashed edge is removed.. In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more ...
The following generalizes Cayley's formula to labelled forests: Let T n,k be the number of labelled forests on n vertices with k connected components, such that vertices 1, 2, ..., k all belong to different connected components. Then T n,k = k n n − k − 1. [5]
The algorithm for weak components generates the strongly connected components in this order, and maintains a partition of the components that have been generated so far into the weak components of their induced subgraph. After all components are generated, this partition will describe the weak components of the whole graph. [2] [3]
In the mathematical field of graph theory, Kirchhoff's theorem or Kirchhoff's matrix tree theorem named after Gustav Kirchhoff is a theorem about the number of spanning trees in a graph, showing that this number can be computed in polynomial time from the determinant of a submatrix of the graph's Laplacian matrix; specifically, the number is equal to any cofactor of the Laplacian matrix.
A graph with three components. In graph theory, a component of an undirected graph is a connected subgraph that is not part of any larger connected subgraph. The components of any graph partition its vertices into disjoint sets, and are the induced subgraphs of those sets. A graph that is itself connected has exactly one component, consisting ...
It is thus identical to the question of the node connectivity of a given graph in discrete mathematics. The vertex-cut version of Menger's theorem also proves that the disconnection number is equivalent to a maximally sized group with a network in which every pair of persons has at least this number of separate paths between them.
The primitive graph operations that the algorithm uses are to enumerate the vertices of the graph, to store data per vertex (if not in the graph data structure itself, then in some table that can use vertices as indices), to enumerate the out-neighbours of a vertex (traverse edges in the forward direction), and to enumerate the in-neighbours of a vertex (traverse edges in the backward ...