The generalized 3-connectivity of Lexicographic product graphs

TL;DR

This paper investigates the generalized 3-connectivity of lexicographic product graphs, establishing lower bounds and sharpness of bounds for these graph products, extending understanding of connectivity in complex networks.

Contribution

It provides new bounds for the generalized 3-connectivity of lexicographic and Cartesian product graphs, including sharp bounds and their proofs.

Findings

Lower bound for $ppa_3(G\u2297 H)$ as $ppa_3(G)|V(H)|$

Upper bounds for $ppa_3(Gox H)$ and $ppa_3(G\u2297 H)$

All bounds proved are sharp

Abstract

The generalized $k$-connectivity $\kappa_k(G)$ of a graph $G$, introduced by Chartrand et al., is a natural and nice generalization of the concept of (vertex-)connectivity. In this paper, we prove that for any two connected graphs $G$ and $H$, $\kappa_3(G\circ H)\geq \kappa_3(G)|V(H)|$. We also give upper bounds for $\kappa_3(G\Box H)$ and $\kappa_3(G\circ H)$. Moreover, all the bounds are sharp.