Related papers: Cayley graphs generated by small degree polynomial…
A new algebraic Cayley graph is constructed using finite fields. Its connectedness and diameter bound are studied via Weil's estimate for character sums. These graphs provide a new source of expander graphs, extending classical results of…
The degree-diameter problem seeks to find the largest possible number of vertices in a graph having given diameter and given maximum degree. Very often the problem is studied for restricted families of graph such as vertex-transitive or…
We show that every non-trivial compact connected group and every non-trivial general or special linear group over an infinite field admits a generating set such that the associated Cayley graph has infinite diameter.
We present simple graph-theoretic characterizations of Cayley graphs for left-cancellative monoids, groups, left-quasigroups and quasigroups. We show that these characterizations are effective for the end-regular graphs of finite degree.
The results of computer searches for large graphs with given (small) degree and diameter are presented. The new graphs are Cayley graphs of semidirect products of cyclic groups and related groups. One fundamental use of our ``dense graphs''…
Let $\Gamma$ be a Cayley graph generated by a transposition tree. A natural problem is to understand how the properties of the Cayley graph depend on those of the underlying transposition tree. We focus here on diameter and distance related…
We provide an index bound for character sums of polynomials over finite fields. This improves the Weil bound for high degree polynomials with small indices, as well as polynomials with large indices that are generated by cyclotomic mappings…
We obtain a new bound connecting the first non--trivial eigenvalue of the Laplace operator of a graph and the diameter of the graph, which is effective for graphs with small diameter or for graphs, having the number of maximal paths…
A directed diameter of a directed graph is the maximum possible distance between a pair of vertices, where paths must respect edge orientations, while undirected diameter is the diameter of the undirected graph obtained by symmetrizing the…
We consider the case in which mixed graphs (with both directed and undirected edges) are Cayley graphs of Abelian groups. In this case, some Moore bounds were derived for the maximum number of vertices that such graphs can attain. We first…
The degree-diameter problem seeks to find the largest possible number of vertices in a graph having given diameter and given maximum degree. There has been much recent interest in the problem for mixed graphs, where we allow both undirected…
In this paper we present a new Good Characterization of maximum genus of a graph which makes a common generalization of the works of Xuong, Liu, and Fu et al. Based on this, we find a new polynomially bounded algorithm to find the maximum…
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.
Paley graphs and Paley sum graphs are classical examples of quasi-random graphs. In this paper, we provide new constructions of families of quasi-random graphs that behave like Paley graphs but are neither Cayley graphs nor Cayley sum…
The unipotent subgroup of a finite group of Lie type over a prime field Z/pZ comes equipped with a natural set of generators; the properties of the Cayley graph associated to this set of generators have been much studied. In the present…
Let $\Gamma$ be a group and $(\Gamma_n)_{n=1} ^{\infty}$ be a descending sequence of finite-index normal subgroups. We establish explicit upper bounds on the diameters of the directed Cayley graphs of the $\Gamma/\Gamma_n$ , under some…
We provide upper bounds for the cardinality of the value set of a polynomial map in several variables over a finite field. These bounds generalize earlier bounds for univariate polynomials.
We prove that the number of iterations taken by the Weisfeiler-Leman algorithm for configurations coming from Schreier graphs is closely linked to the diameter of the graphs themselves: an upper bound is found for general Schreier graphs,…
We present a simple mechanism, which can be randomised, for constructing sparse $3$-uniform hypergraphs with strong expansion properties. These hypergraphs are constructed using Cayley graphs over $\mathbb{Z}_2^t$ and have vertex degree…
Mixed graphs can be seen as digraphs that have both arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case in which such graphs are Cayley graphs of Abelian groups. These groups can be constructed by…