Computing tight upper bounds on the algebraic connectivity of certain graphs

A generalized Bethe tree is a rooted unweighted tree in which vertices at the same level have the same degree. Let B be a generalized Bethe tree. The algebraic connectivity of: the generalized Bethe tree B, a tree obtained from the union of B and a tree J isomorphic to a subtree of B such that the root vertex of J is the root vertex of B. a tree obtained from the union of r generalized Bethe trees joined at their respective root vertices, a graph obtained from the cycle B-r by attaching B, by its root, to each vertex of the cycle, and a tree obtained from the path P-r by attaching B, by its root, to each vertex of the path, is the smallest eigenvalue of a special type of symmetric tridiagonal matrices. In this paper, we first derive a procedure to compute a tight upper bound on the smallest eigenvalue of this special type of matrices. Finally, we apply the procedure to obtain a tight upper bound on the algebraic connectivity of the above mentioned graphs. (C) 2008 Elsevier Inc. All rights reserved.