  An almost self-centered graph is a connected graph of order $n$ with exactly
$n-2$ central vertices, and an almost peripheral graph is a connected graph of
order $n$ with exactly $n-1$ peripheral vertices. We determine (1) the maximum
girth of an almost self-centered graph of order $n;$ (2) the maximum
independence number of an almost self-centered graph of order $n$ and radius
$r;$ (3) the minimum order of a $k$-regular almost self-centered graph and (4)
the maximum size of an almost peripheral graph of order $n;$ (5) which numbers
are possible for the maximum degree of an almost peripheral graph of order $n;$
(6) the maximum number of vertices of maximum degree in an almost peripheral
graph of order $n$ whose maximum degree is the second largest possible.
Whenever the extremal graphs have a neat form, we also describe them.
