The eight 6-vertex asymmetric graphsThe Frucht graph, one of the five smallest asymmetric cubic graphs.
Graph families defined by their automorphisms
distance-transitive
→
distance-regular
←
strongly regular
↓
symmetric (arc-transitive)
←
t-transitive,t ≥ 2
skew-symmetric
↓
(if connected) vertex- and edge-transitive
→
edge-transitive and regular
→
edge-transitive
↓
↓
↓
vertex-transitive
→
regular
→
(if bipartite) biregular
↑
Cayley graph
←
zero-symmetric
asymmetric
In graph theory, a branch of mathematics, an undirected graph is called an asymmetric graph if it has no nontrivial symmetries.
Formally, an automorphism of a graph is a permutation p of its vertices with the property that any two vertices u and v are adjacent if and only if p(u) and p(v) are adjacent.
The identity mapping of a graph onto itself is always an automorphism, and is called the trivial automorphism of the graph. An asymmetric graph is a graph for which there are no other automorphisms.
Note that the term "asymmetric graph" is not a negation of the term "symmetric graph," as the latter refers to a stronger condition than possessing nontrivial symmetries.
In graph theory, a branch of mathematics, an undirected graph is called an asymmetricgraph if it has no nontrivial symmetries. Formally, an automorphism...
undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the...
conventionally the term "symmetric graph" is not complementary to the term "asymmetricgraph," as the latter refers to a graph that has no nontrivial symmetries...
include asymmetric relations, asymmetry of shapes in geometry, asymmetricgraphs et cetera. When determining whether an object is asymmetrical, look for...
automorphisms: An asymmetricgraph is an undirected graph with only the trivial automorphism. A vertex-transitive graph is an undirected graph in which every...
Appendix:Glossary of graph theory in Wiktionary, the free dictionary. This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes...
The adjacency matrix of a directed graph can be asymmetric. One can define the adjacency matrix of a directed graph either such that a non-zero element...
In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group, is a graph that encodes the abstract...
In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix or discrete Laplacian...
product, it can distinguish symmetric and asymmetric facts. This approach is scalable to a large knowledge graph in terms of time and space cost. ANALOGY:...
In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency. A regular...
In graph theory, a strongly regular graph (SRG) is a regular graph G = (V, E) with v vertices and degree k such that for some given integers λ , μ ≥ 0...
yield a TSP problem in asymmetric form. An equivalent formulation in terms of graph theory is: Given a complete weighted graph (where the vertices would...
endorelations. Terminology particular for graph theory is used for description, with an ordinary (undirected) graph presumed to correspond to a symmetric...
Paley graphs form an infinite family of conference graphs, which yield an infinite family of symmetric conference matrices. Paley graphs allow graph-theoretic...
Algebraic graph theory is a branch of mathematics in which algebraic methods are applied to problems about graphs. This is in contrast to geometric, combinatoric...
and only if it is asymmetric. For instance, the Frucht graph has a distinguishing coloring with only one color. In a complete graph, the only distinguishing...
The asymmetric J-curve implies that there could be an asymmetric relationship between the exchange rate changes and trade balance. The asymmetric effects...
In the mathematical field of graph theory, the Rado graph, Erdős–Rényi graph, or random graph is a countably infinite graph that can be constructed (with...
are called asymmetric (or identity) graphs. Frucht's theorem states that any group can be realized as the group of symmetries of a graph, and a strengthening...
closure and transitive reduction are also used in the closely related area of graph theory. A relation R on a set X is transitive if, for all x, y, z in X,...
automatically using graph drawing techniques. In some sources, the phrase "Hasse diagram" has a different meaning: the directed acyclic graph obtained from...
science and network science, network theory is a part of graph theory. It defines networks as graphs where the vertices or edges possess attributes. Network...
Mersenne Twister Coloring algorithm: Graph coloring algorithm. Hopcroft–Karp algorithm: convert a bipartite graph to a maximum cardinality matching Hungarian...
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