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 G_k,l, be the graph obtained from G by attaching two new paths P:vv₁v₂…v_k and Q:vu₁u₂…u_l of length k and l, respectively, at v. We prove that if l?k?1 then μ(G_k-1,l+1)?μ(G_k,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.