Vladimir I. Trofimov

Automorphism Groups of Covering Graphs

For a large class of finite Cayley graphs we construct covering graphs whose automorphism groups coincide with the groups of lifted automorphisms. As an application we present new examples of 1/2-transitive and 1-regular graphs.

Morphology evolution in a growing film

Abstract A general method for quantitative description of growth and morphology evolution in thin film growing via 3D island mechanism is presented. The method is based on the calculation of the survival probability for any point on (or above) a substrate. The growth law for different island growth mode (hemisphere and paraboloid) and condensation regime is derived in a self-consistent manner at all successive deposition stages, which shows growth acceleration at late stages due to island collisions. A temporal evolution of a lateral growth front perimeter that can be used for experimental determination of the growth law is also given. For surface relief, forming at paraboloid growth, the height–height autocorrelation function (ACF) is derived for different hillock shape and different nucleation mode generating different hillock space distribution. It is found that the hillock shape strongly affects the overall surface roughness and the hillock space distribution influences on both the ACF form and the rms roughness. For hemisphere growth surface morphology evolution during film growth is analyzed in terms of the rms surface roughness and the roughness coefficient and it is shown that both of them can be presented as a universal (independent of a growth regime) unimodal function of either coverage or film thickness with a maximum just prior to the completed film formation.

Rate equations model for layer epitaxial growth kinetics

Abstract Recently, we have proposed a simple kinetic model for layer epitaxial growth, which combines the rate equations approach and a feeding zone that allows accounting for the interlayer adatoms diffusion. With this model it has clearly been demonstrated how with decreasing surface adatom diffusivity and/or increasing the repulsive Ehrlich–Schwoebel (ES) barrier height growth, mode crosses over from atomically smooth layer-by-layer growth to a smooth multilayer growth and finally to a rough 3D growth, and corresponding ‘phase diagram’ of the growth mode in parametric space has been constructed. In this paper the role of the growing island collisions in the epitaxial growth is analyzed. To that end the comparative studies of the growth kinetics and morphology evolution in two extreme cases of the island collision process behaviour: impingement and coalescence are performed by numerical integration of the rate equations. It is shown that the character of the island collision process weakly affects on the location of demarcating lines in the phase diagram of the growth mode, which is determined by the growth parameters characterizing the surface adatoms diffusivity (μ) and the ES barrier (ω). It is found that in addition to these growth parameters, there exists important internal parameter, the critical coverage for the next layer nucleation, its magnitude uniquely determines the onset of a growth mode transition irrespective of μ and ω thus remaining constant along a demarcating line. The growing island collisions significantly influence on the nucleation kinetics in successive layers and its scaling with growth parameters μ and ω and corresponding scaling exponents are determined.

Highly arc transitive digraphs: reachability, topological groups

Let D be a locally finite, connected, 1-arc transitive digraph. It is shown that the reachability relation is not universal in D provided that the stabilizer of an edge satisfies certain conditions which seem to be typical for highly arc transitive digraphs. As an implication, the reachability relation cannot be universal in highly arc transitive digraphs with prime in- or out-degree.Two different aspects of the connection between highly arc transitive digraphs and the theory of totally disconnected locally compact groups are also considered.

Automorphism Groups of Graphs with Quadratic Growth

Let?be a graph with almost transitive group Aut(?) and quadratic growth. We show that Aut(?) contains an almost transitive subgroup isomorphic to the free abelian group Z2.

Epitaxial growth kinetics in the presence of an Ehrlich-Schwoebel barrier: comparative analysis of different models

In the last years, much theoretical effort has been devoted to the effects of the Ehrlich–Schwoebel (ES) step-edge barrier to interlayer diffusion on epitaxial growth. In a number of models, in a frame of the rate equation approach the expressions for critical island size for second-layer nucleation have been derived and various growth mode transitions, induced by the ES barrier, have been revealed, and corresponding ‘phase diagrams’ of the growth mode in a parameter space have been constructed. The application of these models to the experimental data has been shown to result in the reasonable estimates of the ES barrier. However, in just published papers by employing kinetic MC simulations and theory, based on a concept of the residence time of an adatom on top of an island and simple statistical arguments, the expressions for critical island size and nucleation rate differing from that of the rate equation approach, have been obtained and it has been remarked that the applicability of mean field rate equations to the confined geometry on top of a small island is not obvious. This article presents a comparative analysis and reconsideration of the rate equation-based models in view of these critical comments that shows that these models are valid at least in a range of weak ES barriers where the most interesting growth phenomena are developed.

Homoepitaxial growth kinetics in the presence of a Schwoebel barrier

Nowadays it is well-recognized that the additional barrier to downhill adatom diffusion at the step edge plays an important role in the epitaxial growth. Very recently we have developed a simple model for homoepitaxial layer growth kinetics which allows to take into account the Schwoebel barrier impact on adatoms interlayer diffusion by using the concept of a feeding zone, as we have proposed earlier. This paper is devoted to further refinement and extension of the model to the cases of an arbitrary nucleus size and coalescence behaviour of growing islands. The model consists of an infinite set of coupled non-linear rate equations for adatom and 2D island surface densities and coverage in each successive growing layer. These equations in combination with an integral condition determining the new layer formation onset fully describe homoepitaxial growth kinetics at predetermined five model parameters, characterizing adatoms diffusion rate, critical nucleus size and stability, Schwoebel barrier effect, and coalescence. The growth mechanisms and kinetics in a wide range of parameter values are studied and growth mechanism phase diagrams in various parameter spaces are constructed and discussed.

Vertex stabilizers of graphs and tracks, I

This paper is devoted to the conjecture saying that, for any connected locally finite graph @C and any vertex-transitive group G of automorphisms of @C, at least one of the following assertions holds: (1) There exists an imprimitivity system @s of G on V(@C) with finite (maybe one-element) blocks such that the stabilizer of a vertex of the factor graph @C/@s in the induced group of automorphisms G^@s is finite. (2) The graph @C is hyperbolic (i.e., for some positive integer n, the graph @C^n defined by V(@C^n)=V(@C) and E(@C^n)={{x,y}:0

On geometric properties of directed vertex-symmetric graphs

Let Γ be a directed locally finite vertex-symmetric graph such that the underlying graph Γ- is connected. We prove that, in the case Γ is infinite, there exists a positive integer k, depending only on min{deg+ (Γ), deg-(Γ)}, such that for any positive integer n there exists a directed path of Γ of length not greater than kn2 whose initial and terminal vertices are at distance n in the graph Γ-. We also prove an analogous result in the case Γ is finite and n < diam(Γ-/2 + 1. The results are deduced from certain isoperimetric type inequalities for cones in Γ.

Undirected and directed graphs with near polynomial growth

The growth function of a graph with respect to a vertex is near polynomial if there exists a polynomial bounding it above for infinitely many positive integers. In the paper vertex-symmetric undirected graphs and vertex-symmetric directed graphs with coinciding inand out-degrees are described in the case their growth functions are near polynomial.