  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 graphs when choosing $E(X,Y) = (Y^2 - f(X))(\lambda Y^2
- f(X))$ with $f(X)$ a polynomial over $\mathbb{F}_q$ and $\lambda$ 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.
