Related papers: Graph kinematics of discrete physical objects: bey…
A concrete analysis of the general properties and numerical characteristics of different atomic and nuclear shell systems and subnuclear particles is carried out on the base of the solution scheme for an introduced in part I physical graph…
In part III is realized the consistent development of Heisenberg--Dyson's two-layer matrix approximation to the graph formalism for postulating discrete physical objects (DPO) introduced in parts I-II in the form of discrete sets of…
Starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the Planck-scale, one of the many problems one has to face is to find the discrete protoforms of…
Quantum geometry on a discrete set means a directed graph with a weight associated to each arrow defining the quantum metric. However, these `lattice spacing' weights do not have to be independent of the direction of the arrow. We use this…
Starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the Planck-scale, one of the many problems one has to face in this enterprise is to find the…
As in an earlier paper we start from the hypothesis that physics on the Planck scale should be described by means of concepts taken from ``discrete mathematics''. This goal is realized by developing a scheme being based on the dynamical…
It is shown that properties of a discrete space-time geometry distinguish from properties of the Riemannian space-time geometry. The discrete geometry is a physical geometry, which is described completely by the world function. The discrete…
The presented paper is a review of papers on the microcosm physics geometrization in the last twenty years. These papers develop a new direction of the microcosm physics. It is so-called geometric paradigm, which is alternative to the…
Here I present a new discrete model of quantum mechanics for relativistic 1-electron systems, in which particle movement is described by a directed space-time graph with attached 4-spinors, but without any continuous wave functions. These…
Rooted, weighted continuum random trees are used to describe limits of sequences of random discrete trees. Formally, they are random quadruples $(\mathcal{T},d,r,p)$, where $(\mathcal{T},d)$ is a tree-like metric space, $r\in\mathcal{T}$ is…
We address the "inverse problem" for discrete geometry, which consists in determining whether, given a discrete structure of a type that does not in general imply geometrical information or even a topology, one can associate with it a…
This is a simple mathematical introduction into Feynman diagram technique, which is a standard physical tool to write perturbative expansions of path integrals near a critical point of the action. I start from a rigorous treatment of a…
A method for considering a weighted directed graph with an accuracy of up to a given partition of the set of vertices is proposed. The resulting digraph (the splitting graph) does not contain arcs inside each partition element, and the arcs…
The threshold-$k$ metric dimension ($\mathrm{Tmd}_k$) of a graph is the minimum number of sensors -- a subset of the vertex set -- needed to uniquely identify any vertex in the graph, solely based on its distances from the sensors, when the…
Starting from the working hypothesis that both physics and the corresponding mathematics and in particular geometry have to be described by means of discrete concepts on the Planck-scale, one of the many problems one has to face in this…
An important application of distance geometry to biochemistry studies the embeddings of the vertices of a weighted graph in the three-dimensional Euclidean space such that the edge weights are equal to the Euclidean distances between…
Introduction: molecular geometry, the three-dimensional arrangement of atoms within a molecule, is fundamental to understanding chemical reactivity, physical properties, and biological activity. The prevailing models used to describe…
We develop a novel method for measuring the similarity between complete weighted graphs, which are probed by means of discrete-time quantum walks. Directly probing complete graphs using discrete-time quantum walks is intractable due to the…
The proper thinness of a graph is an invariant that generalizes the concept of a proper interval graph. Every graph has a numerical value of proper thinness and the graphs with proper thinness~1 are exactly the proper interval graphs. A…
Dynamic graph theory is a novel, growing area that deals with graphs that change over time and is of great utility in modelling modern wireless, mobile and dynamic environments. As a graph evolves, possibly arbitrarily, it is challenging to…