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 (G) and (G \ v), respectively. In this paper, we consider the functions (v) = (G) - (G \ v) and (v) = (G\v) (G) , provide attainable upper and lower bounds on both functions, and characterise the equality cases in those bounds. The function 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

• Type: Journal
• Year: 2010
• Where: DM
• Authors: Steve Kirkland
• Keywords: Algebraic Connectivities, Attainable Upper, Connected Graph, DM 2010