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In the mathematical field of graph theory, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving the edge–vertex connectivity. Formally, an automorphism of a graph G = ( V , E ) is a permutation σ of the vertex set V , such that the pair of vertices ( u , v ) form an edge if and only if ...
The vertical line test, shown graphically. The abscissa shows the domain of the (to be tested) function. In mathematics, the vertical line test is a visual way to determine if a curve is a graph of a function or not. A function can only have one output, y, for each unique input, x.
Integer-distance straight line embeddings are known to exist for cubic graphs. [3] Sachs (1983) raised the question of whether every graph with a linkless embedding in three-dimensional Euclidean space has a linkless embedding in which all edges are represented by straight line segments, analogously to Fáry's theorem for two-dimensional ...
A t-transitive graph is a graph such that the automorphism group acts transitively on t-arcs, but not on (t + 1)-arcs. Since 1-arcs are simply edges, every symmetric graph of degree 3 or more must be t-transitive for some t, and the value of t can be used to further classify symmetric graphs.
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).
Every graph is the line graph of some hypergraph, but, given a fixed edge size k, not every graph is a line graph of some k-uniform hypergraph. A main problem is to characterize those that are, for each k ≥ 3. A hypergraph is linear if each pair of hyperedges intersects in at most one vertex. Every graph is the line graph, not only of some ...
In graph theory, the Weisfeiler Leman graph isomorphism test is a heuristic test for the existence of an isomorphism between two graphs G and H. [1] It is a generalization of the color refinement algorithm and has been first described by Weisfeiler and Leman in 1968. [ 2 ]
The smallest asymmetric regular graphs have ten vertices; there exist 10-vertex asymmetric graphs that are 4-regular and 5-regular. [2] [3] One of the five smallest asymmetric cubic graphs [4] is the twelve-vertex Frucht graph discovered in 1939. [5] According to a strengthened version of Frucht's theorem, there are infinitely many asymmetric ...
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