Related papers: On quadratic orbital networks
Graphs that are critical (minimal excluded minors) for embeddability in surfaces are studied. In Part I we consider the structure of graphs with a 2-vertex-cut that are critical with respect to the Euler genus. A general theorem describing…
We prove that any quantum field theory, or more generally any probability distribution over tempered distributions in $\mathbb{R}^d$, admits a neural network description with a countable infinity of parameters. As an example, we realize the…
The area of networks is very interdisciplinary and exhibits many applications in several fields of science. Nevertheless, there are few studies focusing on geographically located $d$-dimensional networks. In this paper, we study scaling…
We compute the Euler characteristics of tautological vector bundles and their exterior powers over the Quot schemes of curves. We give closed-form expressions over punctual Quot schemes in all genera. For higher rank quotients of a trivial…
The Evasiveness conjecture have been proved for properties of graphs on a prime-power number of vertices and the six vertices case. The ten vertices case is still unsolved. In this paper we study the size of the automorphism group of a…
In this paper we provide a classification of all regular maps on surfaces of Euler characteristic $-r^d$ for some odd prime $r$ and integer $d\ge 1$. Such maps are necessarily non-orientable, and the cases where $d = 1$ or $2$ have been…
The current paper focuses on fundamental groups and Euler characteristics of various digital models of the 2-dimensional sphere. For all models that we consider, we show that the fundamental groups are trivial, and compute the Euler…
Many real-world networks exhibit scale-free feature, have a small diameter and a high clustering tendency. We have studied the properties of a growing network, which has all these features, in which an incoming node is connected to its…
We derive a number of upper and lower bounds for the first nontrivial eigenvalue of a finite quantum graph in terms of the edge connectivity of the graph, i.e., the minimal number of edges which need to be removed to make the graph…
We study the effects of spatial constraints on the structural properties of networks embedded in one or two dimensional space. When nodes are embedded in space, they have a well defined Euclidean distance $r$ between any pair. We assume…
The Weisfeiler-Leman (WL) dimension of a graph is a measure for the inherent descriptive complexity of the graph. While originally derived from a combinatorial graph isomorphism test called the Weisfeiler-Leman algorithm, the WL dimension…
We consider unitary graphs attached to Z_d^n using an analogue of the Euclidean distance. These graphs are shown to be integral when n is odd or the dimension d is even.
The maximum number of vertices in a graph of maximum degree $\Delta\ge 3$ and fixed diameter $k\ge 2$ is upper bounded by $(1+o(1))(\Delta-1)^{k}$. If we restrict our graphs to certain classes, better upper bounds are known. For instance,…
Whitney's theorem states that every 3-connected planar graph is uniquely embeddable on the sphere. On the other hand, it has many inequivalent embeddings on another surface. We shall characterize structures of a $3$-connected $3$-regular…
Universality is one of the key concepts in understanding critical phenomena. However, for interacting inhomogeneous systems described by complex networks a clear understanding of the relevant parameters for universality is still missing.…
Deep connections are known to exist between scale-free networks and non-Gibbsian statistics. For example, typical degree distributions at the thermodynamical limit are of the form $P(k) \propto e_q^{-k/\kappa}$, where the $q$-exponential…
Among all characteristics exhibited by natural and man-made networks the small-world phenomenon is surely the most relevant and popular. But despite its significance, a reliable and comparable quantification of the question `how small is a…
A faithful description of the state of a complex dynamical network would require, in principle, the measurement of all its $d$ variables, an infeasible task for systems with practical limited access and composed of many nodes with high…
Veit Elser proposed a random graph model for percolation in which physical dimension appears as a parameter. Studying this model combinatorially leads naturally to the consideration of numerical graph invariants which we call \emph{Elser…
Many real life networks present an average path length logarithmic with the number of nodes and a degree distribution which follows a power law. Often these networks have also a modular and self-similar structure and, in some cases -…