Quadrance polygons, association schemes and strongly regular graphs

Quadrance between two points A_1 = [x_1,y_1] and A_2 = [x_2,y_2] is the number Q (A_1, A_2) := (x_2 - x_1)^2 + (y_2 - y_1)^2. In this paper, we present some interesting results arise from this notation. In Section 1, we will study geometry over finite fields under quadrance notations. The main purpose of Section 1 is to answer the question, for which a_1,...,a_n, we have a polygon A_1...A_n such that Q(A_i,A_{i+1})=a_i for i = 1,...,n. In Section 2, using tools developed in Section 1, we define a family of association schemes over finite field space F_q x F_q where q is a prime power. These schemes give rise to a graph V_q with vertices the points of F_q^2, and where (X,Y) is an edge of V_q if and only if Q(X,Y) is a nonzero square number in F_q. In Section 3, we will show that V_q is a strongly regular graph and propose a conjecture about the maximal clique of V_q.Comment: 15 pages, submitte

Ramdom walks on hypergroup of circles in finite fields

In this paper we study random walks on the hypergroup of circles in a finite field of prime order p = 4l + 3. We investigating the behavior of random walks on this hypergroup, the equilibrium distribution and the mixing times. We use two different approaches - comparision of Dirichlet forms (geometric bound of eigenvalues), and coupling methods, to show that the mixing time of random walks on hypergroup of circles is only linear.Comment: 14 pages, to appear in Proceeding of Australasian Workshop of Combinatorics Algorithm

Divisor graphs have arbitrary order and size

A divisor graph $G$ is an ordered pair $(V, E)$ where V \subset \mathbbm{Z} and for all $u \neq v \in V$, $u v \in E$ if and only if $u \mid v$ or $v \mid u$. A graph which is isomorphic to a divisor graph is also called a divisor graph. In this note, we will prove that for any $n \geqslant 1$ and $0 \leqslant m \leqslant \binom{n}{2}$ then there exists a divisor graph of order $n$ and size $m$. We also present a simple proof of the characterization of divisor graphs which is due to Chartran, Muntean, Saenpholpant and Zhang.Comment: AWOCA 200

Distribution of determinant of matrices with restricted entries over finite fields

For a prime power $q$, we study the distribution of determinent of matrices with restricted entries over a finite field \mathbbm{F}_q of $q$ elements. More precisely, let $N_d (\mathcal{A}; t)$ be the number of $d \times d$ matrices with entries in $\mathcal{A}$ having determinant $t$. We show that $$N_d (\mathcal{A}; t) = (1 + o (1)) \frac{|\mathcal{A}|^{d^2}}{q},$$ if $|\mathcal{A}| = \omega(q^{\frac{d}{2d-1}})$, $d\geqslant 4$. When $q$ is a prime and $\mathcal{A}$ is a symmetric interval $[-H,H]$, we get the same result for $d\geqslant 3$. This improves a result of Ahmadi and Shparlinski (2007).Comment: Journal of Combinatorics and Number Theory (to appear

The sovability of norm, bilinear and quadratic equations over finite fields via spectra of graphs

In this paper we will give a unified proof of several results on the sovability of systems of certain equations over finite fields, which were recently obtained by Fourier analytic methods. Roughly speaking, we show that almost all systems of norm, bilinear or quadratic equations over finite fields are solvable in any large subset of vector spaces over finite fields.Comment: 28 page

The Erd\"os-Falconer distance problem on the unit sphere in vector spaces over finite fields

art, Iosevich, Koh and Rudnev (2007) show, using Fourier analysis method, that the finite Erd\"os-Falconer distance conjecture holds for subsets of the unit sphere in \mathbbm{F}_q^d. In this note, we give a graph theoretic proof of this result

Some colouring problems for unit-quadrance graphs

The quadrance between two points $A_1 = (x_1, y_1)$ and $A_2 = (x_2, y_2)$ is the number $Q (A_1, A_2) = (x_1 - x_2)^2 + (y_1 - y_2)^2$. Let $q$ be an odd prime power and $F_q$ be the finite field with $q$ elements. The unit-quadrance graph $D_q$ has the vertex set $F_q^2$, and $X, Y \in F_q^2$ are adjacent if and only if $Q (A_1, A_2) = 1$. In this paper, we study some colouring problems for the unit-quadrance graph $D_q$ and discuss some open problems

Random walks on hypergroup of conics in finite fields

In this paper we study random walks on the hypergroup of conics in finite fields. We investigate the behavior of random walks on this hypergroup, the equilibrium distribution and the mixing times. We use the coupling method to show that the mixing time of random walks on hypergroup of conics is only linear.Comment: To appear in the Global Journal in Pure and Applied Mathematic

On chromatic number of unit-quadrance graphs (finite Euclidean graphs)

The quadrance between two points A_1=(x_1, y_1) and A_2=(x_2, y_2) is the number Q (A_1, A_2) = (x_1 - x_2)^2 + (y_1 - y_2)^2. Let q be an odd prime power and F_q be the finite field with $q$ elements. The unit-quadrance graph D_q has the vertex set F_q^2, and X, Y in F_q^2 are adjacent if and only if Q(A_1, A_2) = 1. Let \chi(F_q^2) be the chromatic number of graph D_q. In this note, we will show that q^{1/2}(1/2+o(1)) <= \chi(F_q^2) <= q(1/2 + o(1)). As a corollary, we have a construction of triangle-free graphs D_q of order q^2 with \chi(D_q) >= q/2 for infinitely many values of q.Comment: 5 page

A construction of 3-e.c. graphs using quadrances

A graph is $n$-e.c. ($n$-existentially closed) if for every pair of subsets $A, B$ of vertex set $V$ of the graph such that $A \cap B = \emptyset$ and $|A| + |B| = n$, there is a vertex $z$ not in $A \cup B$ joined to each vertex of $A$ and no vertex of $B$. Few explicit families of $n$-e.c. are known for $n > 2$. In this short note, we give a new construction of 3-e.c. graphs using the notion of quadrance in the finite Euclidean space \mathbbm{Z}_p^d