Related papers: Cayley graphs generated by small degree polynomial…
We present a novel construction of finite groupoids whose Cayley graphs have large girth even w.r.t. a discounted distance measure that contracts arbitrarily long sequences of edges from the same colour class (sub-groupoid), and only counts…
In this paper, we construct some non-normal Cayley graphs and explicitly provide their spectra and eigenspaces using representation theory of finite groups.
In this paper, we derive new sharp diameter bounds for distance regular graphs, which better answer a problem raised by Neumaier and Penji\' c in many cases. Our proof is built upon a relation between the diameter and long-scale Ollivier…
Gcd-graphs over the ring of integers modulo $n$ are a simple and elegant class of integral graphs. The study of these graphs connects multiple areas of mathematics, including graph theory, number theory, and ring theory. In a recent work,…
In this paper, we investigate the existence of fractional revival on Cayley graphs over finite abelian groups. We give a necessary and sufficient condition for Cayley graphs over finite abelian groups to have fractional revival. As…
The $k$-independence number of a graph is the maximum size of a set of vertices at pairwise distance greater than $k$. A graph is called $k$-partially walk-regular if the number of closed walks of a given length $l\le k$, rooted at a vertex…
We raise some questions about graph polynomials, highlighting concepts and phenomena that may merit consideration in the development of a general theory. Our questions are mainly of three types: When do graph polynomials have reduction…
In this paper, we give a new lifting construction of "hyperbolic" type of strongly regular Cayley graphs. Also we give new constructions of strongly regular Cayley graphs over the additive groups of finite fields based on partitions of…
We improve recent results relating graph eigenvalues to other graph parameters like girth, domination number, and minimum degree.
We consider multilinear Littlewood polynomials, polynomials in $n$ variables in which a specified set of monomials $U$ have $\pm 1$ coefficients, and all other coefficients are $0$. We provide upper and lower bounds (which are close for $U$…
In 2006, Collins and Trenk obtained a general sharp upper bound for the distinguishing chromatic number of a connected graph. Inspired by Catlin's combinatorial techniques from 1978, we establish improved upper bounds for classes of…
Several recent papers construct auxiliary polynomials to bound the Weil height of certain classes of algebraic numbers from below. Following these techniques, the author gave a general method for introducing auxiliary polynomials to…
In this paper, we try to determine exact or bounds on the choosability, or list chromatic numbers of some Cayley graphs, typically some Unitary Cayley graphs and Cayley graphs on Dihedral groups.
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 propose a novel construction of finite hypergraphs and relational structures that is based on reduced products with Cayley graphs of groupoids. To this end we construct groupoids whose Cayley graphs have large girth not just in the usual…
Perfect code in Cayley graphs and Cayley sum graphs is studied extensively in recent years. In this paper, we consider perfect code in generalized Cayley graphs.
In this paper, we construct directed strongly regular graphs with new parameters by using partial sum families with local rings. 16 families of new directed strongly regular graphs are obtained and the uniform partial sum families are…
Using the domination parameters of Cayley graphs constructed out of $\mathbb{Z}_{p}\times \mathbb{Z}_{m}$, where $m\in\{p^{\alpha}, p^{\alpha}q^{\beta}, p^{\alpha}q^{\beta}r^{\gamma}\},$ in this paper we are discussing about the total and…
In the present paper, we develop geometric analytic techniques on Cayley graphs of finitely generated abelian groups to study the polynomial growth harmonic functions. We develop a geometric analytic proof of the classical Heilbronn theorem…
We construct a sequence of finite graphs that weakly converge to a Cayley graph, but there is no labelling of the edges that would converge to the corresponding Cayley diagram. A similar construction is used to give graph sequences that…