Let G be a graph in which each vertex initially has weight 1. In each step, the weight from a vertex u to a neighbouring vertex v can be moved, provided that the weight on v is at least as large as the weight on u. The total acquisition number of G, denoted by at(G), is the minimum cardinality of the set of vertices with positive weight at the end of the process. In this paper, we investigate r...

This paper studies the symmetry of random regular graphs and random graphs. Our main result shows that for all 3 ≤ d ≤ n − 4 the random d-regular graph on n vertices almost surely has no non-trivial automorphisms. This answers an open question of N. Wormald [13].

We consider in this note a class of growing random graphs obtained by creating vertices sequentially one by one: at each step, we choose uniformly the neighbours of the newly created vertex; its degree is a random variable with a fixed but arbitrary distribution (depending on the number of existing vertices). Examples from this class turn out to be the Erdős–Rényi random graph, a natural random...

The asymptotic distribution of small subgraphs of a random graph has been basically worked out (see Ruciński [5] for example). But for random regular graphs, the main techniques for proving, for instance, asymptotic normality, do not seem to be usable. One very recent result in this direction is to be found in [3], where switchings were applied to cycle counts. The aim of the present note is to...

In this paper, we study a graph parameter that was recently introduced, the burning number, focusing on a few probabilistic aspects of the problem. The original burning number is revisited and analyzed for binomial random graphs G(n, p), random geometric graphs, and the Cartesian product of paths. Moreover, new variants of the burning number are introduced in which a burning sequence of vertice...

We study a random graph Gn that combines certain aspects of geometric random graphs and preferential attachment graphs. This model yields a graph with power-law degree distribution where the expansion property depends on a tunable parameter of the model. The vertices of Gn are n sequentially generated points x1, x2, . . . , xn chosen uniformly at random from the unit sphere in R. After generati...

Suppose n nodes with n0 acquaintances per node are randomly deployed in a two-dimensional Euclidean space with the geographic restriction that each pair of nodes can exchange information between them directly only if the distance between them is at most r, the acquaintanceship between nodes form a random graph, while the physical communication links constitute a random geometric graph. To get a...

We consider graphs obtained by placing n points at random on a unit sphere in R, and connecting two points by an edge if they are close to each other (e.g., the angle at the origin that their corresponding unit vectors make is at most π/3). We refer to these graphs as geometric graphs. We also consider a complement family of graphs in which two points are connected by an edge if they are far aw...

We introduce and explore random conformal graph directed Markov systems governed by measure-preserving ergodic dynamical systems. We first develop the symbolic thermodynamic formalism for random finitely primitive subshifts of finite type with a countable alphabet (by establishing tightness in a narrow topology). We then construct fibrewise conformal and invariant measures along with fibrewise ...