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

Let G=(V,E) be a tree on ngreater-or-equal, slanted2 vertices and let vset membership, variantV. 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 lgreater-or-equal, slantedkgreater-or-equal, slanted1 then μ(Gk-1,l+1)less-than-or-equals, slantμ(Gk,l). Let (v1,v2) be an edge of G. Let image be the tree obtained from G by deleting the edge (v1,v2) and identifying the vertices v1 and v2. Then we prove that image 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.