A finite group attached to the Laplacian of a graph

Lorenzini, D.J., A finite group attached to the laplacian of a graph, Discrete Mathematics 91 (1991) 277-282. Let F = diag(cp,, . , r~_, , 0), 91, 1 t . 1 q, ~, , denote the Smith normal form of the laplacian matrix associated to a connected graph G on n vertices. Let h denote the cardinal of the set {i 1 rp, > 1). We show that h is bounded by the number of independent cycles of G and we study some cases where these two integers are equal. Let G be a connected graph with m edges, n vertices and adjacency matrix A = (aij). Let dj denote the degree of the ith vertex and define the laplacian of G to be the matrix M:=D-A with D=diag(d,, . . . ,d,). Let ‘J= (1,. . .) 1):Z” -Z. We define @ := Ker ‘JIIm M, where M is thought of as a linear map M: Z”-+ Z”. Let h denote the minima1 number of generators of the group @. Let P(G) = m (n 1) be the number of independent cycles of G. In [2,5.2] we showed that h(G) s P(G). In the present paper, we recall two other descriptions of the group @ and use them to characterize some families of graphs for which the equality h(G) = P(G) holds. We also give a new proof of the inequality L(G) c P(G). The finite abelian group @ can be described in terms of the Smith normal form F=diag(q,, . . . , Q)_~, 0) of M (see [2, 1.41). Any diagonal matrix E = diag(e,, . . . , e,_l, 0), row and column equivalent to M over the integers, induces an isomorphism @ = ZlelZ X . . . X Z/e,_lZ. The integers q1 1 . . . 1 cpn-l can be computed in the following way: vi = Ai/Ai_l where A0 = 1 and Ai is the gcd of the determinants of the i x i minors of M. The Elsevier Science Publishers B.V. (North-Holland)