Let $G$ be a transitive permutation group on finite set $\Omega$ and recall that base for is subset of with trivial pointwise stabiliser. The size $G$, denoted $b(G)$, the minimal base. If $b(G)=2$ then we can study Saxl graph $\Sigma(G)$ which has vertex two vertices are adjacent if they form This vertex-transitive graph, conjectured to connected diameter at most $2$ when primitive. In this pa...

A code C in the Hamming metric, that is, is a subset of vertex set VΓ graph Γ=H(m,q), gives rise to natural distance partition {C,C1,…,Cρ}, where ρ covering radius C. Such called completely transitive if automorphism group Aut(C) acts transitively on each sets C, C1, …, Cρ. 2-neighbour-transitive ρ⩾2 and C1 C2. Let be binary (q=2) having full minimum δ⩾5. Then it known induces 2-homogeneous act...

The random cluster model on a general innnite graph with bounded degree wired at innnity is considered and a \ghost vertex" method is introduced in order to explicitly construct random cluster measures satisfying the Dobrushin-Lanford-Ruelle condition for q 1. It is proved that on a regular nonamenable graph there is a q 0 such that for q q 0 there is a phase transition for an entire interval o...

We consider the class of the topologically locally nite (in short TLF) planar vertex-transitive graphs, a class containing in particular all the one-ended planar Cayley graphs and the normal transitive tilings. We characterize these graphs with a nite local representation and a special kind of nite state automaton named labeling scheme. As a result, we are able to enumerate and describe all TLF...

In this paper, a characterisation is given of finite s-arc transitive Cayley graphs with s ≥ 2. In particular, it is shown that, for any given integer k with k ≥ 3 and k 6= 7, there exists a finite set (maybe empty) of s-transitive Cayley graphs with s ∈ {3, 4, 5, 7} such that all s-transitive Cayley graphs of valency k are their normal covers. This indicates that s-arc transitive Cayley graphs...

A graph G admits an H-covering if every edge of belongs to a subgraph isomorphic given H. is said be H-magic there exists bijection f:V(G)∪E(G)→{1,2,…,|V(G)|+|E(G)|} such that wf(H′)=∑v∈V(H′)f(v)+∑e∈E(H′)f(e) constant, for H′ In particular, H-supermagic f(V(G))={1,2,…,|V(G)|}. When H complete K2, H-(super)magic labeling edge-(super)magic labeling. Suppose F-covering and two graphs F We define (...

We show that the membership problem in a finitely generated submonoid of a graph group (also called a right-angled Artin group or a free partially commutative group) is decidable if and only if the independence graph (commutation graph) is a transitive forest. As a consequence we obtain the first example of a finitely presented group with a decidable generalized word problem that does not have ...