Minimizing algebraic connectivity over connected graphs with fixed girth

Let G,g denote the class of all connected graphs on n vertices with fixed girth g. We prove that if n greater than or equal to 3g-1, then the graph which uniquely minimizes the algebraic connectivity over G,g is the unicyclic "lollipop" graph C-n,C-g obtained by appending a g cycle to a pendant vertex of a path on n - g vertices. The characteristic set of C-n,C-g is also discussed. Throughout both algebraic and combinatorial techniques are used. (C) 2002 Elsevier Science B.V. All rights reserved.