On diagonal equations over finite fields via walks in NEPS of graphs

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Abstract

We obtain an explicit combinatorial formula for the number of solutions (x1,,xr)(Fpab)r to the diagonal equation x1k++xrk=α over the finite field Fpab, with k=pab1b(pa1) and b>1, by using the number of r-walks in NEPS of complete graphs.

Introduction

Diagonal equations  A diagonal equation over the finite field Fpm is an equation of the formα1x1k1++αsxsks=α for αiFpm for i=1,,s and αFpm. This kind of equation has been studied a lot, the interested reader can refer to the pioneering work [14], which relates the number of solutions in term of Gauss sums. Other authors have used Weil's expression to obtain the explicit number of solutions for specific αi's and ki's (see [1], [2], [3], [12], [13], [15], [16]).

In general, it is difficult to find the explicit number of solutions of diagonal equations. In this work we are going to find explicit combinatorial solutions of (1.1) in the finite fields Fpab, when αi=1 and ki=k=pab1b(pa1) for all i and b>0, by using a relation between the number of solutions of (1.1) and the walks of certain graphs which have a special product structure (NEPS).

NEPS operations  Given a set B{0,1}n and graphs G1,,Gn, the NEPS (non-complete extended p-sum) of these graphs with respect to the basis B is the graph G=NEPS(G1,,Gn;B), whose vertex set is the cartesian product of the vertex sets of the individual graphs, V(G)=V(G1)××V(Gn) and two vertices (x1,,xn),(y1,,yn)V(G) are adjacent in G, if and only if there exists some n-tuple (α1,,αn)B such that xi=yi, whenever αi=0, or xi,yi are distinct and adjacent in Gi, whenever αi=1.

These NEPS operations generalize a number of known graph products, all of which have in common that the vertex set of the resulting graph is the cartesian product of the input vertex sets. For instance, NEPS(G1,,Gn;{(1,,1)})=G1Gn is the Kronecker product of the Gi's, NEPS(G1,,Gn;{e1,,en})=G1++Gn (where ei is the vector with 1 in the position i and 0 otherwise) is the sum of the graphs Gi, NEPS(G1,G2;{(1,1),(1,0),(0,1)})=G1G2 is the strong product of G1,G2. In [9], the authors showed that the NEPS of complete graphs are exactly the family of GCD-graphs. We refer to [5] or [6] for the history of the notion of NEPS and to [4], [5] and [8] for readers interested in algebraic graph theory.

Outline and results  The main goal of this paper is to find the number of solutions of the diagonal equationx1k+x2k++xsk=α over finite fields in a combinatorial way. For this purpose, we relate the number of solutions of (1.2) with the number of walks of generalized Paley graphs Γ(k,pm)=Cay(Fq,Rk). By using a classification of generalized Paley graphs which are certain NEPS of complete graphs due to Lim and Praeger (see [10]), the problem of finding the number of solutions of (1.2) turns out to calculate the number of walks of NEPS of complete graphs. We will calculate a closed formula for the number of walks in NEPS of complete graphs.

The paper is organized as follows. In Section 2, we will recall some basic definitions of generalized Paley graphs and diagonal equations over finite fields. Then, we will obtain the direct relationship between the number of r-walks from x to y and the number of solutions of (1.2) with α=yx, in this case.

In Section 3, we will find a closed formula for the number of r-walks in NEPS in terms of the number of walks of its factors, by using some properties of the Kronecker product of matrices and well-known facts about powers of the adjacency matrix and the number of walks in a graph.

In Section 4, we will apply the formula for the number of walks to the case of the cartesian product of the same complete graph which is NEPS(Gn,,Gn;B) with B the canonical basis and Gi=Km for all i=1,n, where Km denotes the complete graph with m vertices. In this case, this graph is the well-known Hamming graph. In [10], the authors characterized those generalized Paley graphs which are Hamming graphs. By using this, we will find the r-walks between two vertices in generalized Paley graphs and thus by applying the result of Section 2 we obtain a formula for the number of solutions of the diagonal equation (1.2) over Fpm for k=pab1b(pa1) where m=ab.

Section snippets

GP-graphs and diagonal equation over finite fields

Let p be a prime and let m,k be positive integers such that k|pm1. The generalized Paley graph (GP-graph for short), is the Cayley graphΓ(k,pm)=Cay(Fpm,Rk)where Rk={xk:xFpm}, i.e. Γ(k,pm) is the graph with set of vertex Fpm and two vertices x,yFpm are neighbors in Γ(k,pm) if and only if yxRk. In general Γ(k,pm) is a directed graph, but if Rk is symmetric (Rk=Rk), then Γ(k,pm) is a simple undirected graph.

Notice that if ω is a primitive element of Fpm, then Rk=ωk, this implies that Γ(k,p

The number of walks in NEPS

It is well-known that if A(G) is the adjacency matrix of a graph G, thenwG(r,vi,vj)=(A(G)r)i,j by labeling the vertices in an appropriate way.

The adjacency matrix of NEPS(G1,,Gn;B) can be calculated in terms of the adjacency matrices of the graphs G1,,Gn. More precisely, if G=NEPS(G1,,Gn;B) and the graphs G1,,Gn have adjacency matrices A1,,An, then the adjacency matrix of G is given byA=αBA1α1Anαn, where ⊗ denotes the Kronecker matrix product and α=(α1,,αn) (see [5]).

Theorem 3.1

If G=NEPS(G1,,Gn

Main results

Let B={e1,,en}, where ei is the n-tuple with 1 in the position i and zeros in the remaining positions. If G is the NEPS of the graphs G1,,Gn with basis B, then G=G1++Gn is the sum of Gt (cartesian product of graph). In this case, we have the following result.

Proposition 4.1

Let G=G1++Gn. Then, we have thatwG(r,vi,vj)=r1++rn=rr!r1!rn!t=1nwGt(rt,πt(vi),πt(vj)), where πt denote the projection of V(G) over V(Gt).

Proof

Let r be a non-negative integer. By Theorem 3.1 we have thatwG(r,vi,vj)=(β1,,βr)Brt=1nwGt(

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Partially supported by CONICET, FonCyT (BID-PICT 2018-02073) and SECyT, UNC (Consolidar 33620180101139CB).

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