  The classical problem of degree sequence realizability asks whether or not a
given sequence of $n$ positive integers is equal to the degree sequence of some
$n$-vertex undirected simple graph. While the realizability problem of degree
sequences has been well studied for different classes of graphs, there has been
relatively little work concerning the realizability of other types of
information profiles, such as the vertex neighborhood profiles.
  In this paper, we initiate the study of neighborhood degree profiles. We
focus on the natural problem of realizing maximum and minimum neighborhood
degrees. More specifically, we ask the following question: Given a sequence $D$
of $n$ non-negative integers $0\leq d_1\leq \cdots \leq d_n$, does there exist
a simple graph with vertices $v_1,\ldots, v_n$ such that for every $1\le i \le
n$, the maximum (resp. minimum) degree in the neighborhood of $v_i$ is exactly
$d_i$?
  We provide in this work various results for both maximum as well as minimum
neighborhood degree for general $n$ vertex graphs. Our results are first of its
kind that studies extremal neighborhood degree profiles. For maximum
neighborhood degree profiles, we provide a {\em complete realizability
criteria}. In comparison, we observe that the minimum neighborhood profiles are
not so well-behaved, for these our necessary and sufficient conditions for
realizability {\em differ by a factor of at most two}.
