All graphs in this paper are nite and undirected with no loops or multiple edges. Let G be a graph. The Laplacian matrix of G is L(G) = D(G)A(G), where D(G) = diag(d(v1); : : : ; d(vn)) is a diagonal matrix and d(v) denotes the degree of the vertex v in G and A(G) is the ad- jacency matrix of G. A one-edge connection of two graphs G1 and G2 is a graph G with V (G) = V (G1) [ V (G2) and E(G) = E(G1) [ E(G2) [ fe = uvg where u 2 V (G1) and v 2 V (G2) is denoted by G = G1#G2. If M is a square matrix, then the minor of the entry in the i-th row and j-th column is the determinant of the submatrix formed by deleting the i-th row and j-th column. This number is often denoted Mi;j . We give a description of the some eigenvalues of L(G = G1#G2) and the relationship between the spectrums of L(G1), L(G2) and L(G = G1#G2), by the Laplacian characteristic polynomial of G.
