Star complements and connectivity in finite graphs

Citation

Rowlinson P (2014) Star complements and connectivity in finite graphs. Linear Algebra and Its Applications, 442, pp. 92-98. https://doi.org/10.1016/j.laa.2013.06.021

Abstract
Let G be a finite graph with H as a star complement for an eigenvalue other than 0 or -1. Let κ(G), δ(G) denote respectively the vertex-connectivity and minimum degree of G. We prove that κ(G) is controlled by δ(G) and κ(H). In particular, for each k∈N there exists a smallest non-negative integer f(k) such that κ(G)⩾k whenever κ(H)⩾k and δ(G)⩾f(k). We show that f(1)=0, f(2)=2, f(3)=3, f(4)=5 and f(5)=7.

Keywords
Graph; Connectivity; Eigenvalue; Star complement

Journal
Linear Algebra and Its Applications: Volume 442