Algebraic connectivity of trees with a pendant edge of infinite weight

Abstract

Let G be a weighted graph. Let v be a vertex of G and let G^v_ω denote the graph obtained by adding a vertex u and an edge {v, u} with weight ω to G. Then the algebraic connectivity µ(G^v_ω) of G^v_ω is a nondecreasing function of ω and is bounded by the algebraic connectivity µ(G) of G. The question of when lim_ω→∞ µ(G^v_ω) is equal to µ(G) is considered and answered in the case that G is a tree.