On fault tolerance of (n,k)-star networks

TL;DR

This paper establishes exact formulas for the fault tolerance measures of $(n,k)$-star networks, specifically the $h$-super connectivity and edge-connectivity, for a certain range of $h$, enhancing understanding of their robustness.

Contribution

It provides the exact values of $h$-super connectivity and edge-connectivity for $(n,k)$-star networks when $n-k \

Findings

Exact formulas for $mbda_s^{(h)}$ and $ppa_s^{(h)}$ for $n-k \

Extends previous results to a larger range of $h$ values.

Confirms the equality of $h$-super connectivity and edge-connectivity for the specified range.

Abstract

Fault tolerance of an $(n,k)$-star network is measured by its $h$-super connectivity $\kappa_s^{(h)}$ or $h$-super edge-connectivity $\lambda_s^{(h)}$. Li {\it et al.} [Appl. Math. Comput. 248 (2014), 525-530; Math. Sci. Lett. 1 (2012), 133-138] determined $\kappa_s^{(h)}$ and $\lambda_s^{(h)}$ for $0\leq h\leq n-k$. This paper determines $\kappa_s^{(h)}=\lambda_s^{(h)}=\frac{(h+1)!(n-h-1)}{(n-k)!}$ for $n-k\leq h \leq n-2$.