Linear Algebra and its Applications 401 (2005) 193–209 www.elsevier.com/locate/laa On distance matrices and Laplacians R. Bapat a,1, S.J. Kirkland b,2,M.Neumannc,∗,3 aIndian Statistical Institute, New Delhi 110016, India bDepartment of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, Canada S4S 0A2 cDepartment of Mathematics, University of Connecticut, Storrs, CT 06269-3009, USA Received 24 September 2003; accepted 19 May 2004 Available online 20 July 2004 Submitted by R. Merris Abstract We consider distance matrices of certain graphs and of points chosen in a rectangular grid. Formulae for the inverse and the determinant of the distance matrix of a weighted tree are obtained. Results concerning the inertia and the determinant of the distance matrix of an unweighted unicyclic graph are proved. If D is the distance matrix of a tree, then we obtain − certain results for a perturbation of D 1.Asanexample,itisshownthatifL is the Laplacian − − 1 matrix of an arbitrary connected graph, then D 1 − L is an entrywise positive matrix. We consider the distance matrix of a subset of a rectangular grid of points in the plane. If we choose m + k − 1 points, not containing a closed path, in an m × k grid, then a formula for the determinant of the distance matrix of such points is obtained. © 2004 Elsevier Inc. All rights reserved. Keywords: Trees; Distance matrices; Laplacians; Determinants; Nonnegative matrices ∗ Corresponding author. E-mail address: [email protected] (M. Neumann). 1 The author thanks the Department of Mathematics, University of Connecticut, for the hospitality provided during the visit when this work was carried out. 2 Research supported in part by NSERC under grant number OGP0138251. 3 The work of this author was supported in part by NSF Grant No. DMS0201333. 0024-3795/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2004.05.011 194 R. Bapat et al. / Linear Algebra and its Applications 401 (2005) 193–209 1. Introduction and background A graph G = (V, E) consists of a finite set of vertices V and a set of edges E. A simple graph has no loops or multiple edges and therefore its edge set consists of distinct pairs. A weighted graph is a graph in which each edge is assigned a weight, which is a positive number. An unweighted graph,orsimplyagraph,is thus a weighted graph with each of the edges bearing weight 1. Let G be a connected, weighted graph on n vertices. The distance between ver- tices i and j is defined to be the minimum weight of all paths from i to j,where the weight of a path is just the sum of the weights of the edges on the path. The distance matrix D of G is an n × n matrix with zeros along the diagonal and with its (i, j)-entry equal to the distance between vertices i and j. Distance matrices of graphs, particularly trees, have been investigated to a great extent in the literature. An early, remarkable result in this context concerns the determinant of the distance matrix of a tree: Graham and Pollack [3] showed that if T is a tree on n vertices with distance matrix D, then the determinant of D is (−1)n−1(n − 1)2n−2, and thus is a function of only the number of vertices; that paper also discusses the inertia of D. (Recall that for symmetric matrix M, its inertia is the triple of integers (n+(M), n0(M), n−(M)),wheren+(M), n0(M),and n−(M) denote the number of positive eigenvalues of M, the multiplicity of 0 as an eigenvalue of M, and the number of negative eigenvalues of M, respectively.) In subsequent work, Graham and Lovasz [4] obtained a formula for D−1, among other results. In Section 2 we extend Graham’s and Lovasz’s formula for D−1 to the case of a weighted tree. We also obtain an extension of the Graham and Pollack determinantal and inertial formulae to the weighted case. In Section 3 we further extend these results to distance matrices arising from unweighted unicyclic graphs. Suppose that we have a weighted graph G = (V, E) with n vertices and m edges, and that we assign an orientation to each edge of G. The associated (vertex-edge) incidence matrix Q of G is the n × m matrix defined as follows. The rows and the columns of Q are indexed by V and E respectively.√ The (i, j)-entry of Q is√ 0 if the ith vertex and the jth edge are not incident and it is w(j) (respectively, − w(j)) if the ith vertex and the jth edge are incident, and the edge originates (respectively, terminates) at the ith vertex, where w(j) denotes the weight of the jth edge. The Laplacian matrix L of G is defined as L = QQT, and is independent of the orienta- tion assigned to G. For basic properties of the Laplacian matrix see [1,7]. We note that in our results involving weighted trees, we will make use of the incidence matrix and the Laplacian matrix that arise by replacing each edge weight of the tree by its reciprocal. In Section 4 we investigate a perturbation problem for distance matrices arising from weighted trees. Let D be a distance matrix arising from a weighted tree and let L be a Laplacian matrix of any weighted graph G.For>0, we consider per- turbations of D−1 of the form D−1 − L and show that matrices of this form are invertible and have a nonnegative inverse. R. Bapat et al. / Linear Algebra and its Applications 401 (2005) 193–209 195 n Recall that if u and v are vectors in R , then the 1-distance between u and − = n | − | v is defined as u v 1 i=1 ui vi . In Section 5 we obtain a formula for the determinant of the 1-distance matrix of a set of points in a rectangular grid. 2 If x1,...,xn are distinct points in R , then their 1-distance matrix D =[di,j ] is an n × n matrix with di,i = 0, i = 1, 2,...,n,anddi,j =xi − xj 1, if i/= j.If m + k − 1 points are chosen from an m × k rectangular grid and if the points do not contain a closed path, then a formula for the determinant of D is obtained. 2. Distance matrix of a tree In this section we extend some well known results on the distance matrix D of an unweighted tree T . The first result is due to Graham and Lovasz [4], who obtained a formula for D−1. The latter two results are due to Graham and Pollack [3], who showed that if T has n vertices, then the determinant of D is (−1)n−1(n − 1)2n−2, and that D has just one positive eigenvalue. In this section, we extend these results to the case of weighted trees. Theorem 2.1. Let T be a weighted tree on n vertices with edge weights α1,...,αn−1 and let D be the corresponding distance matrix. Let L denote the Laplacian matrix for the weighting of T that arises by replacing each edge weight by its reciprocal. For each i = 1,...,n, let di be the degree of the vertex i, let δi = 2 − di, and set T δ =[δ1,...,δn]. Then − 1 1 D 1 =− L + δδT. (2.1) 2 n−1 2 i=1 αi 0 α Proof. We use induction on n.Forn = 2, we have D = 1 , L = α1 0 1 −1 1 1 ,andδ = , and the formula for D−1 follows readily. Now suppose α1 −11 1 we have a weighted tree on n vertices 1, 2,...,n, and form a new weighted tree T on vertices 1,...,n+ 1 by adding in a pendant vertex n + 1, adjacent to vertex n with edge weight αn. Let D, L,andδ be the appropriate quantities for T and let D, ¯ L,andδ be the corresponding quantities for T . Letting en be the nth standard unit basis vector in Rn and 1 be the all ones vector in Rn,wehave + 1 T − 1 − L α enen α en δ en L = n n , δ¯ = , − 1 eT 1 1 αn n αn and D Den + αn1 D = . T + T en D αn1 0 196 R. Bapat et al. / Linear Algebra and its Applications 401 (2005) 193–209 = n−1 = n Let σn−1 i=1 αi and σn i=1 αi, and note that − 1 + 1 ¯ ¯T =−1 + 1 ¯ ¯T L n δδ L δδ 2 2 i=1 αi 2 2σn − 1 − 1 T + 1 T − T − T + T 1 + 1 − 2 L 2α enen 2σ δδ δen enδ enen 2α en 2σ (δ en) = n n n n 1 eT + 1 δT − eT − 1 + 1 2αn n 2σn n 2αn 2σn − 1 + 1 T − T − T − σn−1 T 1 + σn−1 2 L 2σ δδ δen enδ 2α σ enen 2σ δ 2α σ en = n n n n n n . 1 δT + σn−1 eT − σn−1 2σn 2αnσn n 2αnσn −1 1 T 1 From the induction hypothesis, D 1 = δ 1δ = δ,sothatDδ = σn− 1. 2σn−1 σn−1 1 Also from the induction hypothesis, D−1 =−1 L + 1 δδT. We thus find that 2 2σn−1 1 1 − L + δ¯δ¯T 2 2σ n D−1 − αn δδT − 1 δeT + e δT − σn−1 e eT 1 δ + σn−1 e 2σ σ − 2σ n n 2α σ n n 2σ 2α σ n = n n 1 n n n n n n .