In this paper, we generalize the notion of functional graph. Specifically, given an equation $E(X,Y)=0$ with variables X and Y over a finite field ${\mathbb{F}}_q$ of odd characteristic, we define a digraph by choosing the elements in ${\mathbb{F}}_q$ as vertices and drawing an edge from x to y if and only if $E(x,y)=0$. We call this graph as equational graph. In this paper, we study the equational graph when choosing $E(X,Y)=(Y^2-f(X))(\lambda Y^2-f(X))$ with $f(X)$ a polynomial over ${\mathbb{F}}_q$ and λ a non-square element in ${\mathbb{F}}_q$. We show that if f is a permutation polynomial over ${\mathbb{F}}_q$, then every connected component of the graph has a Hamiltonian cycle. Moreover, these Hamiltonian cycles can be used to construct balancing binary sequences. By making computations for permutation polynomials f of low degree, it appears that almost all these graphs are strongly connected, and there are many Hamiltonian cycles in such a graph if it is connected.

在本文中，我们概括了功能图的概念。具体来说，给定一个方程$Ë（X，ÿ）=0$在有限域上具有变量X和Y${\mathbb{F}}_q$ 的奇数特征，我们通过选择元素来定义有向图 ${\mathbb{F}}_q$当且仅当作为顶点并从x到y绘制边线时$Ë（X，ÿ）=0$。我们将此图称为方程图。在本文中，我们研究选择时的方程图$Ë（X，ÿ）=（{ÿ}^2-F（X））（\lambda {ÿ}^2-F（X））$ 与 $F（X）$ 多项式 ${\mathbb{F}}_q$和λ是${\mathbb{F}}_q$。我们证明如果f是一个置换多项式${\mathbb{F}}_q$，则图的每个连接的分量都有一个哈密顿循环。此外，这些哈密顿循环可用于构造平衡二值序列。通过对低次排列的多项式f进行计算，似乎几乎所有这些图都是强连通的，并且如果该图是连通的，则在该图中会有许多哈密顿循环。