On almost self-centered graphs and almost peripheral graphs

TL;DR

This paper investigates extremal properties of almost self-centered and almost peripheral graphs, determining bounds on girth, independence number, degree, and size for various parameters, and characterizing extremal graphs.

Contribution

It provides new bounds and characterizations for extremal properties of almost self-centered and almost peripheral graphs, including girth, independence number, and degree constraints.

Findings

Maximum girth of almost self-centered graphs determined.

Maximum independence number for given radius established.

Bounds on maximum degree and size of almost peripheral graphs derived.

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.