ELA, Volume 13, pp. 175-184, July 2005, abstract.

Algebraic Connectivity of Trees with a Pendant Edge 
of Infinite Weight

Abraham Berman and Karl-Heinz Foerster

Let G be a weighted graph. Let v be a vertex of G and let 
G^v_\omega denote the graph obtained by adding a vertex u 
and an edge {v,u} with weight \omega to G. Then the algebraic 
connectivity \mu(G^v_\omega) of G^v_\omega is a nondecreasing 
function of \omega and is bounded by the algebraic connectivity 
\mu(G) of G. The question of when \lim\mu(G^v_\omega) is equal 
to \mu(G) as \omega tends to infinity is considered and 
answered in the case that G is a tree.