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In mathematics, and, in particular, in graph theory, a rooted graph is a graph in which one vertex has been distinguished as the root. [1] [2] Both directed and undirected versions of rooted graphs have been studied, and there are also variant definitions that allow multiple roots. Examples of rooted graphs with some variants.
If H is a two-vertex complete graph K 2, then for any graph G, the rooted product of G and H has domination number exactly half of its number of vertices. Every connected graph in which the domination number is half the number of vertices arises in this way, with the exception of the four-vertex cycle graph.
A subset of vertices in a graph G is called dissociation if it induces a subgraph with maximum degree 1. distance The distance between any two vertices in a graph is the length of the shortest path having the two vertices as its endpoints. domatic A domatic partition of a graph is a partition of the vertices into dominating sets.
A graph with three vertices and three edges. A graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) [4] [5] is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs {,} of vertices, whose elements are called edges (sometimes links or lines).
The Gram matrix is symmetric in the case the inner product is real-valued; it is Hermitian in the general, complex case by definition of an inner product. The Gram matrix is positive semidefinite, and every positive semidefinite matrix is the Gramian matrix for some set of vectors. The fact that the Gramian matrix is positive-semidefinite can ...
A tree is an undirected graph G that satisfies any of the following equivalent conditions: G is connected and acyclic (contains no cycles). G is acyclic, and a simple cycle is formed if any edge is added to G. G is connected, but would become disconnected if any single edge is removed from G. G is connected and the complete graph K 3 is not a ...
Root Meaning in English Origin language Etymology (root origin) English examples galact-[1] (ΓΛΑΚ) [2] milk: Greek: γάλα, γάλακτος (gála, gálaktos) galactagogue, galactic, galactorrhea, lactose, polygala, polygalactia, galaxy gam-[3] marriage, wedding: Greek
An example for an undirected Graph with a vertex r and its corresponding level structure For the concept in algebraic geometry, see level structure (algebraic geometry) In the mathematical subfield of graph theory a level structure of a rooted graph is a partition of the vertices into subsets that have the same distance from a given root vertex.