Extremizing algebraic connectivity subject to graph theoretic constraints

The main problem of interest is to investigate how the algebraic connectivity o f a weighted connected graph behaves when the graph is perturbed by removing one or more connected components at a xed vertex and replacing this collection by a single connected component. This analysis leads to exhibiting the unique up to isomorphismtrees on n vertices with speciied diameter that maximize and minimize the algebraic connectivity o ver all such trees. When the radius of a graph is the speciied constraint the unique minimizer of the algebraic connectivity o ver all such graphs is also determined. Analogous results are proved for unicyclic graphs with xed girth. In particular, the unique minimizer and maximizer of the algebraic connectivity is given over all such graphs with girth 3.