The ordering of trees and connected graphs by algebraic connectivity Original Research Article

In this paper, we first determine that the first four trees of order ngreater-or-equal, slanted9 with the smallest algebraic connectivity are Pn,Qn,Wn and Zn with α(Pn)<α(Qn)<α(Wn)<α(Zn)<α(T), where T is any tree of order n other than Pn, Qn, Wn, and Zn. Then we consider the effect on the Laplacian eigenvalues of connected graphs by suitably adding edges. By using these results and methods, we finally determine that the first six connected graphs of order ngreater-or-equal, slanted9 with the smallest algebraic connectivity are image and image, with image, where G is any connected graph of order n other than Pn, Qn, image, Wn, image and image.