How to show a graph is vertex-transitive?
A graph in which every edge has the same local environment, so that no edge can be distinguished from any other, is said to be edge-transitive. A undirceted connected graph is edge-transitive if its line graph is vertex-transitive.
What makes a graph transitive?
A graph is vertex-transitive if and only if its graph complement is, since the group actions are identical. Every symmetric graph without isolated vertices is vertex-transitive, and every vertex-transitive graph is regular.
What is automorphism graph theory?
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. That is, it is a graph isomorphism from G to itself.
How do you prove edge transitivity?
In the mathematical field of graph theory, an edge-transitive graph is a graph G such that, given any two edges e1 and e2 of G, there is an automorphism of G that maps e1 to e2. In other words, a graph is edge-transitive if its automorphism group acts transitively on its edges.
How do you determine if a graph is transitive?
An undirected graph has a transitive orientation if its edges can be oriented in such a way that if (x, y) and (y, z) are two edges in the resulting directed graph, there also exists an edge (x, z) in the resulting directed graph.
Is Petersen graph Vertex Transitive?
The Petersen graph is strongly regular (with signature srg(10,3,0,1)). It is also symmetric, meaning that it is edge transitive and vertex transitive.
How do you find the transitivity of a graph?
The transitivity T of a graph is based on the relative number of triangles in the graph, compared to total number of connected triples of nodes. T=3×number of triangles in the networknumber of connected triples of nodes in the network.
What is the degree of a vertex v in a graph?
In graph theory , the degree of a vertex is the number of edges connecting it. In the example below, vertex a has degree 5 , and the rest have degree 1 . A vertex with degree 1 is called an “end vertex” (you can see why).
How do you know if a graph is transitive?
What is the degree of any vertex of graph?
What is a vertex of degree one called?
A vertex with degree 1 is called a leaf vertex or end vertex, and the edge incident with that vertex is called a pendant edge. In the graph on the right, {3,5} is a pendant edge. This terminology is common in the study of trees in graph theory and especially trees as data structures.
What are some examples of vertex-transitive graphs?
The finite Cayley graphs (such as cube-connected cycles) are also vertex-transitive, as are the vertices and edges of the Archimedean solids (though only two of these are symmetric). Potočnik, Spiga and Verret have constructed a census of all connected cubic vertex-transitive graphs on at most 1280 vertices.
When is the vertex-connectivity of a graph equal to D?
If the degree is 4 or less, or the graph is also edge-transitive, or the graph is a minimal Cayley graph, then the vertex-connectivity will also be equal to d. Infinite vertex-transitive graphs include:
What is vertex-transitive automorphism?
such that. In other words, a graph is vertex-transitive if its automorphism group acts transitively upon its vertices. A graph is vertex-transitive if and only if its graph complement is, since the group actions are identical.
How many vertices does a Cayley graph have?
Potočnik, Spiga and Verret have constructed a census of all connected cubic vertex-transitive graphs on at most 1280 vertices. Although every Cayley graph is vertex-transitive, there exist other vertex-transitive graphs that are not Cayley graphs.