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Graphic matroid. In the mathematical theory of matroids, a graphic matroid (also called a cycle matroid or polygon matroid) is a matroid whose independent sets are the forests in a given finite undirected graph. The dual matroids of graphic matroids are called co-graphic matroids or bond matroids. [1] A matroid that is both graphic and co ...
In a given switching class of graphs of a regular two-graph, let Γ x be the unique graph having x as an isolated vertex (this always exists, just take any graph in the class and switch the open neighborhood of x) without the vertex x. That is, the two-graph is the extension of Γ x by x. In the first example above of a regular two-graph, Γ x ...
The Hoffman–Singleton graph. The subgraph of blue edges is a sum of ten disjoint pentagons. In the mathematical field of graph theory, the Hoffman–Singleton graph is a 7- regular undirected graph with 50 vertices and 175 edges. It is the unique strongly regular graph with parameters (50,7,0,1). [5] It was constructed by Alan Hoffman and ...
A 2-vertex-connected graph is series–parallel if and only if it has a nested ear decomposition. Moreover, any open ear decomposition of a 2-vertex-connected series–parallel graph must be nested. The result may be extended to series–parallel graphs that are not 2-vertex-connected by using open ear decompositions that start with a path ...
A 9-vertex graph in which every edge belongs to a unique triangle and every non-edge is the diagonal of a unique quadrilateral. The 99-graph problem asks for a 99-vertex graph with the same property. In graph theory, Conway's 99-graph problem is an unsolved problem asking whether there exists an undirected graph with 99 vertices, in which each ...
The Desargues graph is a symmetric graph: it has symmetries that take any vertex to any other vertex and any edge to any other edge. Its symmetry group has order 240, and is isomorphic to the product of a symmetric group on 5 points with a group of order 2. One can interpret this product representation of the symmetry group in terms of the ...
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