On almost self-centered graphs and almost peripheral graphs

Abstract

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.