Tight bounds on the algebraic connectivity of Bethe trees

A rooted Bethe tree is an unweighted rooted tree of k levels in which the vertex root has degree d, the vertices in level 2 to level (k − 1) have degree (d + 1) and the vertices in level k have degree 1 (pendant vertices). In this paper, we derive tight upper and lower bounds on the algebraic connectivity of (1) a Bethe tree , and (2) a tree obtained by the union of two Bethe trees and having in common the vertex root. A useful tool in our study is the Sherman–Morrison formula.