  Perhaps the very first elementary exercise one encounters in graph theory is
the result that any graph on at least two vertices must have at least two
vertices with the same degree. There are various ways in which this result can
be non-trivially generalised. For example, one can interpret this result as
saying that in any graph $G$ on at least two vertices there is a set $B$ of at
least two vertices such that the difference between the largest and the
smallest degrees (in $G$) of the vertices of $B$ is zero. In this vein we make
the following definition. For any $B\subset V(G)$, let the spread $sp(B)$ of
$B$ be defined to be the difference between the largest and the smallest of the
degrees of the vertices in $B$. For any $k\geq 0$, let $sp(G,k)$ be the largest
cardinality of a set of vertices $B$ such that $sp(B)\leq k$. Therefore the
first elementary result in graph theory says that, for any graph $G$ on at
least two vertices, $sp(G,0)\geq 2$.
  In this paper we first give a proof of a result of Erd\" os, Chen, Rousseau
and Schelp which generalises the above to $sp(G,k)\geq k+2$ for any graph on at
least $k+2$ vertices. Our proof is short and elementary and does not use the
famous Erd\" os-Gallai Theorem on vertex degrees. We then develop lower bounds
for $sp(G,k)$ in terms of the order of $G$ and its minimum, maximum and average
degree. We then use these results to give lower bounds on $sp(G,k)$ for trees
and maximal outerplanar graphs, most of which we show to be sharp.
