Graphs with large generalized (edge-)connectivity

TL;DR

This paper characterizes graphs of order n with specific large generalized (edge-)connectivity values, extending classical connectivity concepts and introducing new characterizations for even k.

Contribution

It provides a characterization of graphs where both generalized k-connectivity and k-edge-connectivity reach specific large values for even k.

Findings

Characterization of graphs with $oxed{ ext{large } ext{generalized } ext{k-} ext{connectivity}}$

Characterization of graphs with $oxed{ ext{large } ext{generalized } ext{k-} ext{edge-connectivity}}$

Results applicable to graph connectivity theory and network reliability

Abstract

The generalized $k$-connectivity $\kappa_k(G)$ of a graph $G$, introduced by Hager in 1985, is a nice generalization of the classical connectivity. Recently, as a natural counterpart, we proposed the concept of generalized $k$-edge-connectivity $\lambda_k(G)$. In this paper, graphs of order $n$ such that $\kappa_k(G)=n-\frac{k}{2}-1$ and $\lambda_k(G)=n-\frac{k}{2}-1$ for even $k$ are characterized.