Trivial Graph: A graph is said to be trivial if a finite graph contains only one vertex and no edge. Simple Graph: A simple graph is a graph which does not contains more than one edge between the pair of vertices.

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 pair (σ(u),σ(v)) also form an edge. That is, it is a graph isomorphism from G to itself.

Order of a graph is the number of vertices in the graph. Size of a graph is the number of edges in the graph. Create some graphs of your own and observe its order and size.

The number of graph edges meeting at a given node in a graph is called the order of that graph vertex.

Remarks. An automorphism is determined by where it sends the generators. An automorphism φ must send generators to generators. In particular, if G is cyclic, then it determines a permutation of the set of (all possible) generators.

In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group.

The order of the matrix is defined as the number of rows and columns. The entries are the numbers in the matrix and each number is known as an element. The size of a matrix is referred to as ‘n by m’ matrix and is written as m×n, where n is the number of rows and m is the number of columns.

The Petersen graph is hypo-Hamiltonian: by deleting any vertex, such as the center vertex in the drawing, the remaining graph is Hamiltonian. This drawing with order-3 symmetry is the one given by Kempe (1886).

Now the generators of Z8 are {ˉ1,ˉ3,ˉ5,ˉ7}, hence Aut(Z8) has order 4. Check that any automorphism f satisfies f2=id, and deduce from this relation that Aut(Z8) is commutative.

The order of a group is the cardinality of its underlying set. In the case of an automorphism group, it is the cardinality of the set of all automorphisms. I.E. (finitely many automorphisms) the number of isomorphisms from a particular group to its self.

The composition of two automorphisms is another automorphism, and the set of automorphisms of a given graph, under the composition operation, forms a group, the automorphism group of the graph. In the opposite direction, by Frucht’s theorem, all groups can be represented as the automorphism group of a connected graph – indeed, of a cubic graph.

In fact, just counting the automorphisms is polynomial-time equivalent to graph isomorphism. This drawing of the Petersen graph displays a subgroup of its symmetries, isomorphic to the dihedral group D5, but the graph has additional symmetries that are not present in the drawing.

For, G and H are isomorphic if and only if the disconnected graph formed by the disjoint union of graphs G and H has an automorphism that swaps the two components. In fact, just counting the automorphisms is polynomial-time equivalent to graph isomorphism.

A distance-transitive graph is a graph such that every pair of vertices may be mapped by an automorphism into any other pair of vertices that are the same distance apart. A semi-symmetric graph is a graph that is edge-transitive but not vertex-transitive.