Diameter and connectivity of finite simple graphs II

TL;DR

This paper investigates the possible combinations of size, free vertices, diameter, and vertex connectivity in finite simple non-complete connected graphs, aiming to classify all feasible sequences of these parameters.

Contribution

It characterizes all integer sequences (n, f, d, k) for which such graphs exist, extending understanding of graph parameter relationships.

Findings

Complete classification of feasible (n, f, d, k) sequences.

Identification of constraints on graph parameters.

Construction methods for graphs with given parameters.

Abstract

Let $G$ be a finite simple non-complete connected graph on $[n] = \{1, \ldots, n\}$ and $\kappa(G) \geq 1$ its vertex connectivity. Let $f(G)$ denote the number of free vertices of $G$ and $\mathrm{diam}(G)$ the diameter of $G$. The final goal of this paper is to determine all sequences of integers $(n,f,d,k)$ with $n\geq 8$, $f\geq 0$, $d\geq 2$ and $k\geq 1$ for which there exists a finite simple non-complete connected graph on $[n]$ with $f=f(G)$, $d=\mathrm{diam}(G)$ and $k=\kappa(G)$.