Related papers: Fractal squares with finitely many connected compo…
We say that a graph is intrinsically non-trivial if every spatial embedding of the graph contains a non-trivial spatial subgraph. We prove that an intrinsically non-trivial graph is intrinsically linked, namely every spatial embedding of…
For a connected graph G=(V,E), a subset U of V is called a disconnected cut if U disconnects the graph and the subgraph induced by U is disconnected as well. We show that the problem to test whether a graph has a disconnected cut is…
Conformal mapping, a classical topic in complex analysis and differential geometry, has become a subject of great interest in the area of surface parameterization in recent decades with various applications in science and engineering.…
Hilbert-space fragmentation provides a mechanism for ergodicity breaking in quantum many-body systems even in the absence of disorder, leading to dynamically disconnected sectors and strong memory of initial conditions. However, identifying…
Introduced the quantitative measure of the structural complexity of the graph (complex network, etc.) based on a procedure similar to the renormalization process, considering the difference between actual and averaged graph structures on…
The boundaries of central place models proved to be fractal lines, which compose fractal texture of central place networks. A textural fractal can be employed to explain the scale-free property of regional boundaries such as border lines,…
We propose a mapping from fracture systems consisting of intersecting fracture sheets in three dimensions to an abstract network consisting of nodes and links. This makes it possible to analyze fracture systems with the methods developed…
A $3$-connected graph $G$ is essentially $4$-connected if, for any $3$-cut $S\subseteq V(G)$ of $G$, at most one component of $G-S$ contains at least two vertices. We prove that every essentially $4$-connected maximal planar graph $G$ on…
We find that the fractal scaling in a class of scale-free networks originates from the underlying tree structure called skeleton, a special type of spanning tree based on the edge betweenness centrality. The fractal skeleton has the…
Covering a network with the minimum possible number of boxes can reveal interesting features for the network structure, especially in terms of self-similar or fractal characteristics. Considerable attention has been recently devoted to this…
The foundation of the theory presented here has already been proved to be effective for the case of curves belonging to the Koch family. The present paper extends the investigation to more complex curves, namely randomly generated curves…
One of the most well known random fractals is the so-called Fractal percolation set. This is defined as follows: we divide the unique cube in $\mathbb{R}^d$ into $M^d$ congruent sub-cubes. For each of these cubes a certain retention…
The purpose of the present paper is to present the main applications of a new method for the determination of the fractal structure of plane curves. It is focused on the inverse problem, that is, given a curve in the plane, find its fractal…
We consider the classification of fractal square dendrites $K$ based on the types of the self-similar boundary $\partial K$ and the main tree $\gamma$ of such dendrites. We show that the self-similar boundary of a fractal square dendrite…
We report an algorithm for the partition of a line segment according to a given ratio $\nu$. At each step the length distribution among sets of the partition follows a binomial distribution. We call $k$-set to the set of elements with the…
The square of a graph is obtained by adding additional edges joining all pair of vertices of distance two in the original graph. Particularly, if $C$ is a hamiltonian cycle of a graph $G$, then the square of $C$ is called a hamiltonian…
While finite graphs have tree-decompositions that efficiently distinguish all their tangles, locally finite graphs with thick ends need not have such tree-decompositions. We show that every locally finite graph without thick ends admits…
Traffic networks have been proved to be fractal systems. However, previous studies mainly focused on monofractal networks, while complex systems are of multifractal structure. This paper is devoted to exploring the general regularities of…
We study derivations and Fredholm modules on metric spaces with a local regular conservative Dirichlet form. In particular, on finitely ramified fractals, we show that there is a non-trivial Fredholm module if and only if the fractal is not…
Frequently, knots are enumerated by their crossing number. However, the number of knots with crossing number $c$ grows exponentially with $c$, and to date computer-assisted proofs can only classify diagrams up to around twenty crossings.…