Effective equidistribution of primitive rational points on expanding horospheres

We prove an effective version of a result due to Einsiedler, Mozes, Shah and Shapira who established the equidistribution of primitive rational points on expanding horospheres in the space of unimodular lattices in at least $3$ dimensions. Their proof uses techniques from homogeneous dynamics and relies in particular on measure-classification theorems --- an approach which does not lend itself to effective bounds. We implement a strategy based on spectral theory, Fourier analysis and Weil's bound for Kloosterman sums in order to quantify the rate of equidistribution for a specific horospherical subgroup in any dimension. We apply our result to provide a rate of convergence to the limiting distribution for the appropriately rescaled diameters of random circulant graphs.Comment: 21 pages, incorporates the referee's comments and correction

Algebraic and Computer-based Methods in the Undirected Degree/diameter Problem - a Brief Survey

This paper discusses the most popular algebraic techniques and computational methods that have been used to construct large graphs with given degree and diameter

Diameters of random Cayley graphs of finite nilpotent groups

We prove the existence of a limiting distribution for the appropriately rescaled diameters of random undirected Cayley graphs of finite nilpotent groups of bounded rank and nilpotency class, thus extending a result of Shapira and Zuck which dealt with the case of abelian groups. The limiting distribution is defined on a space of unimodular lattices, as in the case of random Cayley graphs of abelian groups. Our result, when specialised to a certain family of unitriangular groups, establishes a very recent conjecture of Hermon and Thomas. We derive this as a consequence of a general inequality, showing that the diameter of a Cayley graph of a nilpotent group is governed by the diameter of its abelianisation

Diameter, Girth And Other Properties Of Highly Symmetric Graphs

We consider a number of problems in graph theory, with the unifying theme being the properties of graphs which have a high degree of symmetry. In the degree-diameter problem, we consider the question of finding asymptotically large graphs of given degree and diameter. We improve a number of the current best published results in the case of Cayley graphs of cyclic, dihedral and general groups. In the degree-diameter problem for mixed graphs, we give a new corrected formula for the Moore bound and show non-existence of mixed Cayley graphs of diameter 2 attaining the Moore bound for a range of open cases. In the degree-girth problem, we investigate the graphs of Lazebnik, Ustimenko and Woldar which are the best asymptotic family identified to date. We give new information on the automorphism groups of these graphs, and show that they are more highly symmetrical than has been known to date. We study a related problem in group theory concerning product-free sets in groups, and in particular those groups whose maximal product-free subsets are complete. We take a large step towards a classification of such groups, and find an application to the degree-diameter problem which allows us to improve an asymptotic bound for diameter 2 Cayley graphs of elementary abelian groups. Finally, we study the problem of graphs embedded on surfaces where the induced map is regular and has an automorphism group in a particular family. We give a complete enumeration of all such maps and study their properties

Comparing Two Thickened Cycles: A Generalization of Spectral Inequalities

Motivated by an effort to simplify the Watts-Strogatz model for small-world networks, we generalize a theorem concerning interlacing inequalities for the eigenvalues of the normalized Laplacians of two graphs differing by a single edge. Our generalization allows weighted edges and certain instances of self loops. These inequalities were first proved by Chen et. al in [2] but our argument generalizes the simplified argument given by Li in [8]