then $F$ is called as *s-plateaued* following the terminology in [@mos1].

The set $G_{F}=\{(x,F(x)):x\in \mathbb{F}_{p^{n}}\}$ is called the graph of $F$. Next we will characterize vectorial functions by their graphs.

\[plateaued-PGDS\] Let $F:\mathbb{F}_{p^{n}} \rightarrow \mathbb{F}_{p^{m}}$ be a vectorial function. Then the graph of $F$ is a $(p^{n+m},p^n; \alpha,\beta)$ partial geometric difference set in $H=\mathbb{F}_{p^{n}}\times\mathbb{F}_{p^m}$ satisfying $\beta-\alpha=\theta$ if and only if $|\widehat{F_b}(a)| \in \{0, \sqrt{\theta}\}$ for all non zero $b \in \mathbb{F}_{p^{m}}$ and $a \in \mathbb{F}_{p^{n}}$. In particular, $\alpha=p^{2n-m}-p^{n+s-m}$ and $\beta=p^{n+s}+p^{2n-m}-p^{n+s-m}$.

A non-principal character of $\mathbb{F}_{p^{n}}\times \mathbb{F}_{p^{m}}$ can be written as $\chi_{(a,b)}(x,y)=\zeta_p^{Tr_{n}(ax)+Tr_m(by)}$ for a nonzero $(a,b) \in \mathbb{F}_{p^{n}}\times \mathbb{F}_{p^{m}}$. For any nonzero $b \in {{\mathbb F}}_{p^n}$, the Walsh transform of $F_b$ is $$\begin{aligned}
    \widehat{F_b}(a)=&\sum_{x\in \mathbb{F}_{p^{n}}}\zeta_p^{-Tr_{n}(ax)+Tr_m(bF(x))}=\sum_{x\in \mathbb{F}_{p^{n}}}\chi_{(-a,b)}(x,F_b(x)) \\
    =&\chi_{(-a,b)}(G_F)
    \end{aligned}$$ for any $a \in F_{p^m}$. Therefore $|\widehat{F_b}(a)| \in \{0, \sqrt{\theta}\} |$ for all non zero $b \in \mathbb{F}_{p^{m}}$ if and only if $G_{F}=\{(x,F(x)):x\in \mathbb{F}_{p^{n}}\}$ is a $(p^{n+m},p^n;\alpha,\beta)$ partial geometric difference set satisfying $\beta-\alpha=\theta$. Using the well-known *Parseval identiy*, one immediately sees that $\theta=p^{n+s}$ and $|\{a\in \mathbb{F}_{p^{n}}:\widehat{F_b}(a)\neq 0\}|=p^{n-s}$ for some $0 \leq s \leq n$. The parameters of a partial geometric difference set satisfies the relation in [@olmez1] and hence we have $$p^{3n}=(\beta-\alpha)p^n+\alpha\nu=p^{n+s}p^n+\alpha\nu.$$ Then we see that $\alpha=p^{2n-m}-p^{n+s-m}$ and $\beta=p^{n+s}+p^{2n-m}-p^{n+s-m}$.

Theorem \[plateaued-PGDS\] implies 