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

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. AMS Subject Classifications: 05C50, 15A18, 15A42