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(For a graph with n vertices and r terminals, they use t = n − r − 1 added vertices per tree.) Then, they ask for the k -minimum spanning tree in this augmented graph with k = rt . The only way to include this many vertices in a k -spanning tree is to use at least one vertex from each added tree, for there are not enough vertices remaining ...
A claw-free graph is a graph in which no induced subgraph is a claw; i.e., any subset of four vertices has other than only three edges connecting them in this pattern. Equivalently, a claw-free graph is a graph in which the neighborhood of any vertex is the complement of a triangle-free graph.
From the handshaking lemma, a k-regular graph with odd k has an even number of vertices. A theorem by Nash-Williams says that every k ‑regular graph on 2k + 1 vertices has a Hamiltonian cycle. Let A be the adjacency matrix of a graph. Then the graph is regular if and only if = (, …,) is an eigenvector of A. [2]
A complete graph (denoted , where is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum possible degree, . In a signed graph , the number of positive edges connected to the vertex v {\displaystyle v} is called positive deg ( v ) {\displaystyle (v)} and the number of connected negative ...
The degree distribution is very important in studying both real networks, such as the Internet and social networks, and theoretical networks.The simplest network model, for example, the (Erdős–Rényi model) random graph, in which each of n nodes is independently connected (or not) with probability p (or 1 − p), has a binomial distribution of degrees k:
A complete graph with n nodes represents the edges of an (n – 1)-simplex. Geometrically K 3 forms the edge set of a triangle, K 4 a tetrahedron, etc. The Császár polyhedron, a nonconvex polyhedron with the topology of a torus, has the complete graph K 7 as its skeleton. [15] Every neighborly polytope in four or more dimensions also has a ...
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