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For each natural number n, the edgeless graph (or empty graph) K n of order n is the graph with n vertices and zero edges. An edgeless graph is occasionally referred to as a null graph in contexts where the order-zero graph is not permitted. [1] [2] It is a 0-regular graph. The notation K n arises from the fact that the n-vertex edgeless graph ...
The complement graph of a complete graph is an empty graph. If the edges of a complete graph are each given an orientation, the resulting directed graph is called a tournament. K n can be decomposed into n trees T i such that T i has i vertices. [6] Ringel's conjecture asks if the complete graph K 2n+1 can be decomposed into copies of any tree ...
A path graph or linear graph of order n ≥ 2 is a graph in which the vertices can be listed in an order v 1, v 2, …, v n such that the edges are the {v i, v i+1} where i = 1, 2, …, n − 1. Path graphs can be characterized as connected graphs in which the degree of all but two vertices is 2 and the degree of the two remaining vertices is 1.
In the language of graph theory, the Ramsey number is the minimum number of vertices, v = R(m, n), such that all undirected simple graphs of order v, contain a clique of order m, or an independent set of order n. Ramsey's theorem states that such a number exists for all m and n. By symmetry, it is true that R(m, n) = R(n, m).
The nullity of a graph in the mathematical subject of graph theory can mean either of two unrelated numbers. If the graph has n vertices and m edges, then: In the matrix theory of graphs, the nullity of the graph is the nullity of the adjacency matrix A of the graph. The nullity of A is given by n − r where r is the rank of the adjacency
This sortable list points to the articles describing various individual (finite) graphs. [1] The columns 'vertices', 'edges', ' radius ', ' diameter ', ' girth ', 'P' (whether the graph is planar ), χ ( chromatic number ) and χ' ( chromatic index ) are also sortable, allowing to search for a parameter or another.
Equivalently, the rank of a graph is the rank of the oriented incidence matrix associated with the graph. [2] Analogously, the nullity of the graph is the nullity of its oriented incidence matrix, given by the formula m − n + c, where n and c are as above and m is the number of edges in the graph. The nullity is equal to the first Betti ...
1. A universal graph is a graph that contains as subgraphs all graphs in a given family of graphs, or all graphs of a given size or order within a given family of graphs. 2. A universal vertex (also called an apex or dominating vertex) is a vertex that is adjacent to every other vertex in the graph.