No Starlike Trees Are Laplacian Cospectral

Let G be a graph with n vertices and m edges. The degree sequence of G is denoted by d1 ≥ d2 ≥ · · · ≥ dn. Let A(G) and D(G) = diag(di : 1 ≤ i ≤ n) be the adjacency matrix and the degree diagonal matrix of G, respectively. The Laplacian matrix of G is L(G) = D(G) − A(G). It is well known that L(G) is a symmetric, semidefinite matrix. We assume the spectrum of L(G), or the Laplacian spectrum of G, is λ = λ1 ≥ λ2 ≥ · · · ≥ λn = 0. If more than one graph is involved, we may write λi(G) in place of λi. λn−1 is called the algebraic connectivity of G and λn−1 > 0 if and only if G is connected. The multiplicity of zero as an eigenvalue equals to the number of components of G. The characteristic polynomial of L(G) can be written by