The notion of stationary Apollonian packings in the d-dimensional Euclidean space is introduced as a mathematical formalization of so-called random Apollonian packings and rotational random Apollonian packings, which constitute popular grain packing models in physics. Apart from dealing with issues of existence and uniqueness in the entire Euclidean space, asymptotic results are provided for th...

In this paper a subset of High-Dimensional Random Apollonian networks, that we called Wheel Random Apollonian Graphs (WRAG), is considered. We show how to generate a Wheel Random Apollonian Graph from a wheel graph. We analyse some basic graph properties like vertices and edges cardinality, some question concerning cycles and the chromaticity in such type of graphs, we suggest further work on t...

A generalization of Apollonian packing of circles Gerhard Guettler Colin Mallowsy Abstract Three circles touching one another at distinct points form two curvilinear triangles. Into one of these we can pack three new circles, touching each other, with each new circle touching two of the original circles. In such a sextuple of circles there are three pairs of circles, with each of the circles in...

In this paper we find an exact analytical expression for the number of spanning trees in Apollonian networks. This parameter can be related to significant topological and dynamic properties of the networks, including percolation, epidemic spreading, synchronization, and random walks. As Apollonian networks constitute an interesting family of maximal planar graphs which are simultaneously small-...

We extend the common theme of random Apollonian packing of circles to consider orientable grains with a noncircular shape. Systems of up to 10(6) grains are examined for a range of polygonal and elliptical shapes using both the random Apollonian packing model and the new rotational random Apollonian packing model which takes into account the extra rotational degree of freedom of noncircular gra...

The curvatures of the circles in integral Apollonian circle packings, named for Apollonius of Perga (262-190 BC), form an infinite collection of integers whose Diophantine properties have recently seen a surge in interest. Here, we give a new description of Apollonian circle packings built upon the study of the collection of bases of Z[i], inspired by, and intimately related to, the ‘sensual qu...

We investigate the majority-vote model with two states (-1,+1) and a noise parameter q on Apollonian networks. The main result found here is the presence of the phase transition as a function of the noise parameter q. Previous results on the Ising model in Apollonian networks have reported no presence of a phase transition. We also studied the effect of redirecting a fraction p of the links of ...