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Two vertices (or edges) of a periodic graph are symmetric if they are in the same orbit of the symmetry group of the graph; in other words, two vertices (or edges) are symmetric if there is a symmetry of the net that moves one onto the other. In chemistry, there is a tendency to refer to orbits of vertices or edges as “kinds” of vertices or ...
Multiple edges, not allowed under the definition above, are two or more edges with both the same tail and the same head. In one more general sense of the term allowing multiple edges, [5] a directed graph is an ordered triple = (,,) comprising: , a set of vertices (also called nodes or points);
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).
where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Euler's polyhedron formula. Thus the number of vertices is 2 more than the excess of the number of edges over the number of faces. For example, since a cube has 12 edges and 6 faces, the formula implies that it has eight vertices.
This hypergraph has order 7 and size 4. Here, edges do not just connect two vertices but several, and are represented by colors. Alternative representation of the hypergraph reported in the figure above, called PAOH. [1] Edges are vertical lines connecting vertices. V7 is an isolated vertex. Vertices are aligned to the left.
When modelling relations between two different classes of objects, bipartite graphs very often arise naturally. For instance, a graph of football players and clubs, with an edge between a player and a club if the player has played for that club, is a natural example of an affiliation network, a type of bipartite graph used in social network analysis.
The skeleton of the tetrahedron (comprising the vertices and edges) forms a graph, with 4 vertices, and 6 edges. It is a special case of the complete graph , K 4 , and wheel graph , W 4 . [ 48 ] It is one of 5 Platonic graphs , each a skeleton of its Platonic solid .
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 with n edges. [7] This is known to be true for sufficiently ...