Related papers: Constructions of Strongly Regular Cayley Graphs Us…
In this paper we introduce and study a type of Cayley graph -- subnormal Cayley graph. We prove that a subnormal 2-arc transitive Cayley graph is a normal Cayley graph or a normal cover of a complete bipartite graph $K_{p^d,p^d}$ with $p$…
In this paper, we construct intriguing sets in five classes of strongly regular graphs defined on nonisotropic points of finite classical polar spaces, and determine their intersection numbers.
We investigate subsets of the symmetric group with structure similar to that of a graph. The trees of these subsets correspond to minimal conjugate generating sets of the symmetric group. There are two main theorems in this paper. The first…
We give a new proof of a theorem of D. Calegari that says that the Cayley graph of a surface group with respect to any generating set lying in finitely many mapping class group orbits has infinite diameter. This applies, for instance, to…
A strongly regular graph is called trivial if it or its complement is a union of disjoint cliques. We prove that every infinite family of nontrivial strongly regular graphs is quasi-random in the sense of Chung, Graham and Wilson.
The concept of directed strongly regular graphs was introduced by Duval in his paper, A Directed Graph Version of Strongly Regular Graphs. Duval also provided several construction methods for directed strongly regular graphs. The directed…
We are interested in various aspects of spectral rigidity of Cayley and Schreier graphs of finitely generated groups. For each pair of integers $d\geq 2$ and $m \ge 1$, we consider an uncountable family of groups of automorphisms of the…
Let $k$ be a number field. We refine a construction of Mestre--Shioda to construct (infinite) families of hyperelliptic curves $X/{k}$ having a record number of rational points and record Mordell--Weil rank relative to the genus of $g$ of…
In this paper a wide family of identifying codes over regular Cayley graphs of degree four which are built over finite Abelian groups is presented. Some of the codes in this construction are also perfect. The graphs considered include some…
In this paper, we formally introduce the concept of a row-sum matrix over an arbitrary group $G$. When $G$ is cyclic, these types of matrices have been widely used to build uniform 2-factorizations of small Cayley graphs (or, Cayley…
We give variants of the Krein bound and the absolute bound for graphs with a spectrum similar to that of a strongly regular graph. In particular, we investigate what we call approximately strongly regular graphs. We apply our results to…
We provide an abundance of strongly regular graphs (SRGs) for certain parameters $(n, k, \lambda, \mu)$ with $n < 100$. For this we use Godsil-McKay (GM) switching with a partition of type $4,n-4$ and Wang-Qiu-Hu (WQH) switching with a…
In this paper, firstly, we provide some necessary and sufficient conditions for generalized Cayley graphs on abelian groups to be bipartite. Secondly, we deduce several necessary and sufficient conditions for generalized Cayley graphs on…
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…
We construct an infinite family of triples (G,S1, S2) each consisting of a group G and a pair (S1, S2) of distinct subsets of G with the following properties. i The two Cayley graphs Cay(G, S1) and Cay(G,S2) are non-isomorphic. ii The…
We find closed form formulas for the Kemeny's constant and the Kirchhoff index for the cluster $G_1\{G_2\}$ of two highly symmetric graphs $G_1$, $G_2$, in terms of the parameters of the original graphs. We also discuss some necessary…
This document is an expanded version of a lecture presented at a conference on "Thin Groups and Superstrong Approximation" held at the Mathematical Sciences Research Institute in February 2012. Superstrong approximation is a criterion on a…
This article lays the foundations for an analogue of geometric group theory that studies actions on graphs by right quasigroups, including racks and quandles. We study markings of graphs that realize racks, and we introduce (di)graph…
A survey is given on mathematical structures which emerge in multi-loop Feynman diagrams. These are multiply nested sums, and, associated to them by an inverse Mellin transform, specific iterated integrals. Both classes lead to sets of…
We define standardized constructions of finite fields, and standardized generators of (multiplicative) cyclic subgroups in these fields. The motivation is to provide a substitute for Conway polynomials which can be used by various software…