Diameter and connectivity of finite simple graphs

TL;DR

This paper characterizes and classifies finite simple connected graphs based on their diameter, vertex connectivity, and number of free vertices, especially those satisfying a specific relation involving these parameters.

Contribution

It determines the possible parameter sequences and classifies graphs that satisfy the relation f + diam = n + 2 - κ, advancing understanding of graph structure.

Findings

Characterization of sequences (n, q, f, d) for such graphs.

Classification of graphs satisfying f + diam = n + 2 - κ.

Insights into the structure related to the depth of binomial edge ideals.

Abstract

Let $G$ be a finite simple non-complete connected graph on $\{1, \ldots, n\}$ 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 $G$. Being motivated by the computation of the depth of the binomial edge ideal of $G$, the possible sequences $(n, q, f, d)$ of integers for which there is a finite simple non-complete connected graph $G$ on $\{1, \ldots, n\}$ with $q = \kappa(G), f = f(G), d = \mathrm{diam}(G)$ satisfying $f + d = n + 2 - q$ will be determined. Furthermore, finite simple non-complete connected graphs $G$ on $\{1, \ldots, n\}$ satisfying $f(G) + \mathrm{diam}(G) = n + 2 - \kappa(G)$ will be classified.