In the graph theory, two graphs are said to be isomorphic if there is a one-one, onto mapping defined between their set of vertices so as to preserving the adjacency between vertices. An isomorphism defined on a vertex set of a graph to itself is called automorphism of the given graph. Two vertices in a graph are said to be similar if there is an automorphism defined on its vertex set mapping o...

We show that almost every Cayley graph Γ of an abelian group G of odd prime-power order has automorphism group as small as possible. Additionally, we show that almost every Cayley (di)graph Γ of an abelian group G of odd prime-power order that does not have automorphism group as small as possible is a normal Cayley (di)graph of G (that is, GL/Aut(Γ)).

We examine designs and binary codes associated with the line graph of the n-cube Qn, i.e. the Hamming graph H(n, 2). We find the automorphism groups and the parameters of the codes. We find a regular subgroup of the automorphism group that can be used for permutation decoding, or partial permutation decoding, for any information set.

The automorphism groups Aut(C(G, X)) and Aut(CM(G, X, p)) of a Cayley graph C(G, X) and a Cayley map CM(G, X, p) both contain an isomorphic copy of the underlying group G acting via left translations. In our paper, we show that both automorphism groups are rotary extensions of the group G by the stabilizer subgroup of the vertex 1G . We use this description to derive necessary and sufficient co...

The automorphic A-chromatic index of a graph Γ is the minimum integer m for which Γ has a proper edge-coloring with m colors which is preserved by a given subgroup A of the full automorphism group of Γ. We compute the automorphic A-chromatic index of each generalized Petersen graph when A is the full automorphism group.

A pile-scramble shuffle is one of the most effective shuffles in card-based cryptography. Indeed, many protocols are constructed from shuffles. This paper aims to study power In particular, for any directed graph G, we introduce a new protocol called “a G”, and show that it can be implemented using only. Our proposed requires $$2(n+m)$$ cards, where n m numbers vertices edges respectively. The ...

The notions of symbolic matrix system and λ-graph system for a subshift are generalizations of symbolic matrix and λ-graph (= finite symbolic matrix) for a sofic shift respectively ([Doc. Math. 4(1999), 285-340]). M. Nasu introduced the notion of textile system for a pair of graph homomorphisms to study automorphisms and endomorphisms of topological Markov shifts ([Mem. Amer. Math. Soc. 546,114...

In this paper we use orderly algorithms to enumerate (perfect) one-factorizations of complete graphs, the automorphism groups of which contain certain prescribed subgroups. We showed that, for the complete graph Ki2, excluding those one-factorizations containing exactly one automorphism of six disjoint cycles of length two, there are precisely 56391 nonisomorphic one-factorizations of Ki2 with ...