The effect on the algebraic connectivity of a tree by grafting or collapsing of edges

Let G=(V,E) be a tree on n⩾2 vertices and let v∈V. Let L(G) be the Laplacian matrix of G and μ(G) be its algebraic connectivity. Let Gk,l, be the graph obtained from G by attaching two new paths P:vv1v2…vk and Q:vu1u2…ul of length k and l, respectively, at v. We prove that if l⩾k⩾1 then μ(Gk-1,l+1)⩽μ(Gk,l). Let (v1,v2) be an edge of G. Let be the tree obtained from G by deleting the edge (v1,v2) and identifying the vertices v1 and v2. Then we prove that As a corollary to the above results, we obtain the celebrated theorem on algebraic connectivity which states that among all trees on n vertices, the path has the smallest and the star has the largest algebraic connectivity.