On diagonal equations over finite fields via walks in NEPS of graphs☆
Introduction
Diagonal equations A diagonal equation over the finite field is an equation of the form for for and . 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 's and '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 , when and for all i and , 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 and graphs , the NEPS (non-complete extended p-sum) of these graphs with respect to the basis is the graph , whose vertex set is the cartesian product of the vertex sets of the individual graphs, and two vertices are adjacent in G, if and only if there exists some n-tuple such that , whenever , or are distinct and adjacent in , whenever .
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, is the Kronecker product of the 's, (where is the vector with 1 in the position i and 0 otherwise) is the sum of the graphs , is the strong product of . 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 equation 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 . 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 , 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 with the canonical basis and for all , where 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 for where .
Section snippets
GP-graphs and diagonal equation over finite fields
Let p be a prime and let be positive integers such that . The generalized Paley graph (GP-graph for short), is the Cayley graph i.e. is the graph with set of vertex and two vertices are neighbors in if and only if . In general is a directed graph, but if is symmetric (), then is a simple undirected graph.
Notice that if ω is a primitive element of , then , this implies that
The number of walks in NEPS
It is well-known that if is the adjacency matrix of a graph G, then by labeling the vertices in an appropriate way.
The adjacency matrix of can be calculated in terms of the adjacency matrices of the graphs . More precisely, if and the graphs have adjacency matrices , then the adjacency matrix of G is given by where ⊗ denotes the Kronecker matrix product and (see [5]).
Theorem 3.1 If
Main results
Let , where is the n-tuple with 1 in the position i and zeros in the remaining positions. If G is the NEPS of the graphs with basis , then is the sum of (cartesian product of graph). In this case, we have the following result. Proposition 4.1 Let . Then, we have that where denote the projection of over . Proof Let r be a non-negative integer. By Theorem 3.1 we have that
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Partially supported by CONICET, FonCyT (BID-PICT 2018-02073) and SECyT, UNC (Consolidar 33620180101139CB).