Narrow Results

A heap ordered tree of size n is a planted plane tree together with a bijection from the nodes to the set {1,…,n} which is monotonically increasing when going from the root to the leaves. In a recent paper by Chen and Ni, the expectation and the variance of the depth of a random node in a random heap ordered tree of size n was considered. Here, we give additional results concerning level polynomials. From a historical point of view, we trace these trees back to an earlier paper by Prodinger and Urbanek and find formulae that are proved in the paper by Chen and Ni by ad hoc computations already in a classic book by Greene and Knuth. Also, we mention that a paper by Bergeron, Flajolet and Salvy develops a more general theory of increasing trees, based on simply generated families of trees. Furthermore we consider the path length which is a natural concept when dealing with the depth. Expectation and variance are obtained, both explicitly and asymptotically.

This paper clarifies the relation between synchronization and graph topology. Applying the Connection Graph Stability method developed by Belykh et al. [2004a] to the study of synchronization in networks of coupled oscillators, we show which graph properties matter for synchronization. In particular, while we explicitly link the stability of synchronization with the average path length for a wide class of coupling graphs, we prove by a simple argument that the average path length is not always the crucial quantity for synchronization. We also show that synchronization in scale-free networks can be described by means of regular networks with a star-like coupling structure. Finally, by considering an example of coupled Hindmarsh–Rose neuron models, we demonstrate how global stability of synchronization depends on the parameters of the individual oscillator.

We define and study a class of fractal dendrites called triangular labyrinth fractals. For the construction, we use triangular labyrinth pattern systems, consisting of two triangular patterns: a white and a yellow one. Correspondingly, we have two fractals: a white and a yellow one. The fractals studied here are self-similar, and fit into the framework of graph directed constructions. The main results consist in showing how special families of triangular labyrinth patterns systems, which are defined based on some shape features, can generate exactly three types of dendrites: labyrinth fractals where all nontrivial arcs have infinite length, fractals where all nontrivial arcs have finite length, or fractals where the only arcs of finite lengths are line segments parallel to a certain direction. We also study the existence of tangents to arcs. The paper is inspired by research done on labyrinth fractals in the unit square that have been studied during the last decade. In the triangular case, due to the geometry of triangular shapes, some new techniques and ideas are necessary in order to obtain the results.

We introduce and study mixed triangular labyrinthic fractals, which can be seen as an extension of (generalized) Sierpiński gaskets. This is a new class of fractal dendrites that generalize the self-similar triangular labyrinth fractals studied recently by the authors. A mixed triangular labyrinthic fractal is defined by a triangular labyrinthic pattern and an infinite sequence of triangular labyrinth patterns systems, whereas one triangular labyrinth patterns system is sufficient for generating a self-similar triangular labyrinth fractal. Moreover, a labyrinthic pattern is more general than a labyrinth pattern, and the idea is that the use of many different patterns provides objects with a “richer” structure. After proving that these non-self-similar fractals are dendrites, we study the growth of path lengths in graphs associated with iterations. We prove geometric properties of the fractals, some of which have rather “local” character and on the other hand occur at sites distributed “all over” the fractals.