A circulant (di)graph is a (di)graph on n vertices that admits a cyclic automorphism of order n. This paper provides a survey of the work that has been done on finding the automorphism groups of circulant (di)graphs, including the generalisation in which the edges of the (di)graph have been assigned colours that are invariant under the aforementioned cyclic automorphism. Mathematics Subject Cla...

Constructing symmetric drawings of graphs is NP-hard. In this paper, we present a new method for drawing graphs symmetrically based on group theory. More formally, we define an n-geometric automorphism group as a subgroup of the automorphism group of a graph that can be displayed as symmetries of a drawing of the graph in n dimensions. Then we present an algorithm to find all 2and 3-geometric a...

ABSTRACT Let ‎G=(V,E) ‎be a‎ ‎simple ‎connected ‎graph ‎with ‎vertex ‎set ‎V‎‎‎ ‎and ‎edge ‎set ‎‎‎E. ‎The Szeged index ‎of ‎‎G is defined by ‎ where ‎ respectively ‎ ‎ is the number of vertices of ‎G ‎closer to ‎u‎ (‎‎respectively v)‎ ‎‎than ‎‎‎v (‎‎respectively u‎).‎ ‎‎If ‎‎‎‎S ‎is a‎ ‎set ‎of ‎size‎ ‎ ‎ ‎let ‎‎V ‎be ‎the ‎set ‎of ‎all ‎subsets ‎of ‎‎S ‎of ‎size ‎3. ‎Then ‎we ‎define ‎t...

The modified bubble-sort graph of dimension n is the Cayley graph of Sn generated by n cyclically adjacent transpositions. In the present paper, it is shown that the automorphism group of the modified bubble sort graph of dimension n is Sn×D2n, for all n ≥ 5. Thus, a complete structural description of the automorphism group of the modified bubble-sort graph is obtained. A similar direct product...

In this paper we shall present a natural generalisation of the notion of automorphism of a graph or digraph G, namely a two-fold automorphism. This is a pair (α, β) of permutations of the vertex set V(G) which acts on ordered pairs of vertices of G in the natural way. The action of (α, β) on all such ordered pairs gives a graph which is two-fold isomorphic to G. When α = β the two-fold automorp...

A tricirculant is a graph admitting a non-identity automorphism having three cycles of equal length in its cycle decomposition. A graph is said to be symmetric if its automorphism group acts transitively on the set of its arcs. In this paper it is shown that the complete bipartite graph K3,3, the Pappus graph, Tutte’s 8-cage and the unique cubic symmetric graph of order 54 are the only connecte...

Effective search for graph automorphisms allows identifying symmetries in many discrete structures, ranging from chemical molecules to microprocessor circuits. Using this type of structure can enhance visualization as well as speed up computational optimization and verification. Competitive algorithms for the graph automorphism problem are based on efficient partition refinement augmented with ...

The augmented cube graph AQn is the Cayley graph of Z n 2 with respect to the set of 2n− 1 generators {e1, e2, . . . , en, 00 . . . 0011, 00 . . . 0111, 11 . . . 1111}. It is known that the order of the automorphism group of the graph AQn is 2 n+3, for all n ≥ 4. In the present paper, we obtain the structure of the automorphism

Hodges et al. showed that the countable random graph has the small index property. The stronger result of the title is deduced from this and a general theorem about permutation groups. A consequence is that the automorphism group of the random graph is not isomorphic to the automorphism group of any other countable homogeneous graph or digraph.