Related papers: Infinite Paley graphs
We define a growing model of random graphs. Given a sequence of nonnegative integers $\{d_n\}_{n=0}^\infty$ with the property that $d_i\leq i$, we construct a random graph on countably infinitely many vertices $v_0,v_1\ldots$ by the…
The Erd\H{o}s, Gr\"unwald, and Weiszfeld theorem is a characterization of those infinite graphs which are Eulerian. That is, infinite graphs that admit infinite Eulerian paths. In this article we prove an effective version of the Erd\H{o}s,…
We introduce a new class of countably infinite random geometric graphs, whose vertices are points in a metric space, and vertices are adjacent independently with probability p if the metric distance between the vertices is below a given…
A characterization is given of finite groups $H$ that have skew-morphisms of order coprime to the order $|H|$, and their skew-morphisms. A complete classification is then given of the automorphism groups and the underlying graphs of…
This work will appear as a chapter in a forthcoming volume titled "Topics in Probabilistic Graph Theory". A theory of scaling limits for random graphs has been developed in recent years. This theory gives access to the large-scale geometric…
Given a finite field, one can form a directed graph using the field elements as vertices and connecting two vertices if their difference lies in a fixed subgroup of the multiplicative group. If -1 is contained in this fixed subgroup, then…
We study covering graphs of the Paley graph associated to a finite field of characteristic p in the case where the covering transformation group is cyclic of prime order distinct from p. When the field has q = p elements, we show that the…
We show the maximality of subfields as cliques in a special family of Cayley graphs defined on the additive group of a finite field. In particular, this confirms a conjecture of Yip on generalized Paley graphs.
We prove a sampling theorem for infinite-dimensional Paley-Wiener spaces on graphs which allows for stable frame reconstruction. We prove that all sampling sets for a fixed Paley-Wiener space are complements of lambda-sets (i.e. sets where…
A graph is called integral if its eigenvalues are integers. In this article, we provide the necessary and sufficient conditions for a Cayley graph over a finite symmetric algebra $R$ to be integral. This generalizes the work of So who…
It is proved that with finitely many possible exceptions, each cyclotomic scheme over finite field is determined up to isomorphism by the tensor of 2-dimensional intersection numbers; for infinitely many schemes, this result cannot be…
We extend the Weil representation of infinite-dimensional symplectic group to a representation a certain category of linear relations.
We describe automorphisms and derivations of several important associative and Lie algebras of infinite matrices over a field.
For each infinite word over a given finite alphabet, we define an increasing sequence of rooted finite graphs, that can be thought as approximations of the famous Sierpinski carpet. These sequences naturally converge to an infinite rooted…
We describe some metric properties of incomparability graphs. We consider the problem of the existence of infinite paths, either induced or isometric, in the incomparability graph of a poset. Among other things, we show that if the…
We consider the infinite-dimensional hypercube graph. This graph is not connected and has isomorphic connected components. We describe the restrictions of its automorphisms to the connected components and the automorphism group of connected…
We introduce a new graph invariant of finite groups that provides a complete characterization of the splitting types of unramified prime ideals in normal number field extensions entirely in terms of the Galois group. In particular, each…
We describe the central measures for the random walk on graded graphs. Using this description, we obtain the list of all finite traces on three infinite-dimensional algebras: on the Brauer algebra, on the partition algebra, and on the…
Recently, variants of many classical extremal theorems have been proved in the random environment. We, complementing existing results, extend the Erd\H{o}s-Gallai Theorem in random graphs. In particular, we determine, up to a constant…
A graph is Cartesian decomposable if it is isomorphic to a Cartesian product of (more than one) strictly smaller graphs, each of which has more than one vertex and admits no such decomposition. These smaller graphs are called the…