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In an undirected simple graph of order n, the maximum degree of each vertex is n − 1 and the maximum size of the graph is n(n − 1) / 2 . The edges of an undirected simple graph permitting loops induce a symmetric homogeneous relation on the vertices of that is called the adjacency relation of .
Whenever two edges of a Euclidean minimum spanning tree meet at a vertex, they must form an angle of 60° or more, with equality only when they form two sides of an equilateral triangle. This is because, for two edges forming any sharper angle, one of the two edges could be replaced by the third, shorter edge of the triangle they form, forming ...
Applying the Mycielskian repeatedly, starting with the one-edge graph, produces a sequence of graphs M i = μ(M i−1), sometimes called the Mycielski graphs. The first few graphs in this sequence are the graph M 2 = K 2 with two vertices connected by an edge, the cycle graph M 3 = C 5 , and the Grötzsch graph M 4 with 11 vertices and 20 edges.
A useful graph that is often associated with a triangulation of a polygon P is the dual graph. Given a triangulation T P of P , one defines the graph G ( T P ) as the graph whose vertex set are the triangles of T P , two vertices (triangles) being adjacent if and only if they share a diagonal.
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. [1] [2] Such a drawing is called a plane graph, or a planar embedding of the graph.
A book, book graph, or triangular book is a complete tripartite graph K 1,1,n; a collection of n triangles joined at a shared edge. 2. Another type of graph, also called a book, or a quadrilateral book, is a collection of 4 -cycles joined at a shared edge; the Cartesian product of a star with an edge.
A hypergraph is a combinatorial structure that, like an undirected graph, has vertices and edges, but in which the edges may be arbitrary sets of vertices rather than having to have exactly two endpoints. A bipartite graph (,,) may be used to model a hypergraph in which U is the set of vertices of the hypergraph, V is the set of hyperedges, and ...
In the mathematical discipline of graph theory, the line graph of an undirected graph G is another graph L(G) that represents the adjacencies between edges of G. L(G) is constructed in the following way: for each edge in G, make a vertex in L(G); for every two edges in G that have a vertex in common, make an edge between their corresponding vertices in L(G).
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