Algebraic connectivity for vertex-deleted subgraphs, and a notion of vertex centrality

Abstract

Let G be a connected graph, suppose that v is a vertex of G , and denote the subgraph formed from G by deleting vertex v by G ∖ v . Denote the algebraic connectivities of G and G ∖ v by α ( G ) and α ( G ∖ v ) , respectively. In this paper, we consider the functions ϕ ( v ) = α ( G ) − α ( G ∖ v ) and κ ( v ) = α ( G ∖ v ) α ( G ) , provide attainable upper and lower bounds on both functions, and characterise the equality cases in those bounds. The function κ yields a measure of vertex centrality, and we apply that measure to analyse certain graphs arising from food webs.