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The graph of any cubic function is similar to such a curve. The graph of a cubic function is a cubic curve, though many cubic curves are not graphs of functions. Although cubic functions depend on four parameters, their graph can have only very few shapes. In fact, the graph of a cubic function is always similar to the graph of a function of ...
According to Brooks' theorem every connected cubic graph other than the complete graph K 4 has a vertex coloring with at most three colors. Therefore, every connected cubic graph other than K 4 has an independent set of at least n/3 vertices, where n is the number of vertices in the graph: for instance, the largest color class in a 3-coloring has at least this many vertices.
In graph theory, the hypercube graph Q n is the graph formed from the vertices and edges of an n-dimensional hypercube. For instance, the cube graph Q 3 is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube. Q n has 2 n vertices, 2 n – 1 n edges, and is a regular graph with n edges touching each vertex.
A key theorem in the theory of word-representable graphs states that a graph is word-representable iff it admits a semi-transitive orientation. [7] As a corollary to the proof of the key theorem one obtain an upper bound on word-representants: Each non-complete word-representable graph G is 2( n − κ ( G ))-representable, where κ ( G ) is ...
The cubical graph is a special case of hypercube graph or -cube—denoted as —because it can be constructed by using the operation known as the Cartesian product of graphs. To put it in a plain, its construction involves two graphs connecting the pair of vertices with an edge to form a new graph. [ 30 ]
The halved cube graph of dimension four includes all of the cube vertices and edges, and all of the edges of the two demicubes. In graph theory, the halved cube graph or half cube graph of dimension n is the graph of the demihypercube, formed by connecting pairs of vertices at distance exactly two from each other in the hypercube graph.
Many important graph families can be characterized in terms of the induced paths or cycles of the graphs in the family. Trivially, the connected graphs with no induced path of length two are the complete graphs, and the connected graphs with no induced cycle are the trees. A triangle-free graph is a graph with no induced cycle of length three.
An ordered pair of vertices, such as an edge in a directed graph. An arrow (x, y) has a tail x, a head y, and a direction from x to y; y is said to be the direct successor to x and x the direct predecessor to y. The arrow (y, x) is the inverted arrow of the arrow (x, y). articulation point A vertex in a connected graph whose removal would ...