On Almost Self-centered Graphs and Almost Peripheral Graphs

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