Generalized Connectivity of Star Graphs

TL;DR

This paper establishes a generalized connectivity measure for star graphs, determining the minimum vertices or edges removal needed to disconnect the graph without vertices of degree less than a specified value, confirming a prior conjecture.

Contribution

It provides a new generalized connectivity formula for star graphs, confirming Wan and Zhang's conjecture and extending understanding of their fault tolerance properties.

Findings

Minimum removal of vertices or edges is quantified as at least (k+1)!(n-k-1).

The result confirms Wan and Zhang's conjecture on star graph connectivity.

The connectivity measure applies for all integers 0 ≤ k ≤ n-2.

Abstract

This paper shows that, for any integers $n$ and $k$ with $0\leqslant k \leqslant n-2$, at least $(k+1)!(n-k-1)$ vertices or edges have to be removed from an $n$-dimensional star graph to make it disconnected and no vertices of degree less than $k$. The result gives an affirmative answer to the conjecture proposed by Wan and Zhang [Applied Mathematics Letters, 22 (2009), 264-267].