On Almost Self-centered Graphs and Almost Peripheral Graphs

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.