Extremizing algebraic connectivity subject to graph theoretic constraints

Abstract

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