characteristic, the graph of $f:\mathbb{F}_{p^n}\rightarrow \mathbb{F}_{p^m}$, $G_{f}=\{(x,f(x)):x\in \mathbb{F}_{p^{n}}\}$, plays an important role for the relation to difference sets, [@pott1; @tan2010]. For instance, the graph of a $p$-ary bent function can be recognized as a relative difference set. In general, a characterization of plateaued functions in terms of difference sets is not known. A partial result in this direction is provided in [@cmt] for partially bent functions which is a subfamily of plateaued functions.

There are recent result concerning explicit characterization of plateaued functions in odd characteristics through their second order derivatives in [@CMOS; @mos1; @mos2].

In this paper, we first investigate the link between the graph of a plateaued function and partial geometric difference set. We also provide several characterization of plateaued functions in terms of associated difference set properties. By using these characterization we provide a family of vectorial plateaued functions which has an interesting connection to three-valued cross correlation functions.

The organization of the paper is as follows. In Section 2, we provide preliminary results concerning partial geometric difference sets. In Section 3, we mainly provide the links between vectorial plateaued functions and partial geometric difference sets. We also provide a construction as a result of our characterizations. 