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 produ...

Interval graphs are intersection graphs of closed intervals and circle graphs are intersection graphs of chords of a circle. We study automorphism groups of these graphs. We show that interval graphs have the same automorphism groups as trees, and circle graphs have the same as pseudoforests, which are graphs with at most one cycle in every connected component. Our technique determines automorp...

Let S be a set of transpositions such that the girth of the transposition graph of S is at least 5. It is shown that the automorphism group of the Cayley graph of the permutation group H generated by S is the semidirect product R(H) ⋊ Aut(H,S), where R(H) is the right regular representation of H and Aut(H,S) is the set of automorphisms of H that fixes S setwise. Furthermore, if the connected co...

An automorphism α of a Cayley graph Cay(G,S) of a group G with connection set S is color-preserving if α(g, gs) = (h, hs) or (h, hs−1) for every edge (g, gs) ∈ E(Cay(G,S)). If every color-preserving automorphism of Cay(G,S) is also affine, then Cay(G,S) is a CCA (Cayley color automorphism) graph. If every Cayley graph Cay(G,S) is a CCA graph, then G is a CCA group. Hujdurović, Kutnar, D.W. Morr...

Any finite group can be encoded as the automorphism group of an unlabeled simple graph. Recently Hartke, Kolb, Nishikawa, and Stolee (2010) demonstrated a construction that allows any ordered pair of finite groups to be represented as the automorphism group of a graph and a vertexdeleted subgraph. In this note, we describe a generalized scenario as a game between a player and an adversary: an a...

The commuting graph ∆(G) of a group G is the graph whose vertex set is the group and two distinct elements x and y being adjacent if and only if xy = yx. In this paper the automorphism group of this graph is investigated. We observe that Aut(∆(G)) is a non-abelian group such that its order is not prime power and square-free.

A distinguishing colouring of a graph is a colouring of the vertex set such that no non-trivial automorphism preserves the colouring. Tucker conjectured that if every non-trivial automorphism of a locally finite graph moves infinitely many vertices, then there is a distinguishing 2-colouring. We show that the requirement of local finiteness is necessary by giving a nonlocally finite graph for w...

A regular and edge-transitive graph which is not vertex-transitive is said to be semisymmetric. Every semisymmetric graph is necessarily bipartite, with the two parts having equal size and the automorphism group acting transitively on each of these parts. A semisymmetric graph is called biprimitive if its automorphism group acts primitively on each part. In this paper biprimitive graphs of smal...

A graph is said to be 1 2-transitive if its automorphism group is ver-tex and edge but not arc-transitive. For each n 11, a 1 2-transitive graph of valency 4 and girth 6, with the automorphism group isomor-phic to A n Z 2 , is given.

Any countable Kn-free graph T embeds as a moiety into the universal homogeneous Kn-free graph Kn in such a way that every automorphism of T extends to a unique automorphism of Kn. Furthermore, there are 2 such embeddings which are pairwise not conjugate under Aut(Kn).