Linear and Multilinear Alebra Vol. 59, No. 2, December 20, 393 397 Inverse of the distance matrix of a block rah R.B. Baat a and Sivaramakrishnan Sivasubramanian b * a Stat-Math Unit, Indian Statistical Institute, Delhi, 7-SJSS Mar, New Delhi 0 06, India; b Deartment of Mathematics, Indian Institute of Technoloy, Mumbai 400 076, India Communicated by S. Kirkland (Received 28 May 2009; final version received 9 January 20) A connected rah, whose 2-connected blocks are all cliques (of ossibly varyin sizes) is called a block rah. Let D be its distance matrix. By a theorem of raham, Hoffman and Hosoya, we have det(d) 6¼ 0. We ive a formula for both the determinant and the inverse, D of D. Keywords: distance matrix; determinant; Lalacian matrix AMS Subject Classifications: 5A09; 5A5; 5A24. Introduction raham et al. [3] roved a very attractive theorem about the determinant of the distance matrix D of a connected rah as a function of the distance matrix of its 2-connected blocks. In a connected rah, the distance between two vertices d(u, v) is the lenth of the shortest ath between them. Let A be an n n matrix. Recall that for i, j n, the cofactor c i,j is defined as ( ) iþj times the determinant of the submatrix obtained by deletin row i and column j of A. For a matrix A, let #(A) ¼ P i,jc i,j be the sum of its cofactors. raham et al. [3] showed the followin theorem. THEOREM If is a connected rah with 2-connected blocks, 2,..., r, then #ðd Þ¼ Q r i¼ #ðd i Þ and detðd Þ¼ P r i¼ detðd i Þ Q j6¼i #ðd j Þ. Let D be the distance matrix of a connected rah, all of whose blocks are cliques. Such rahs are called block rahs in [2] and let i denote the blocks of (for i r). See Fiure for an examle. Further, we ive a formula for det(d) for the distance matrix D of a block rah in terms of its block sizes and n, its number of vertices. From the formula it will be clear that det(d) 6¼ 0. Hence, we are interested in findin D. For the case when all blocks are K 2 s (i.e. the rah is a tree) it is known [,4] that D ¼ L 2 þ 2ðn Þ t, where L is the Lalacian matrix of and is the n column vector with v ¼ 2 de v. Similarly, it is known that when all blocks are K 3 s [5], we have D ¼ L 3 þ 3ðn Þ t where L is aain the Lalacian of and *Corresondin author. Email: krishnan@math.iitb.ac.in ISSN 0308 087 rint/issn 563 539 online ß 20 Taylor & Francis htt://dx.doi.or/0.080/0308087.20.557374 htt://www.tandfonline.com
394 R.B. Baat and S. Sivasubramanian 3 4 2 5 9 6 8 7 0 Fiure. An examle of a block rah. is the column vector with v ¼ 3 de v. Thus, D is a constant times L lus a multile of a rank one matrix. We show a similar statement for block rahs. 2. Determinant and inverse of D Let be a block rah on n vertices with blocks i, i r, where each i is a i -clique. Denote by the non-zero constant ¼ Xr The followin theorem is easily derived from Theorem. i¼ i i : ðþ THEOREM 2 Let be a block rah on n vertices with blocks i,ir, where each i is a i -clique. Let D be its distance matrix. Then, detðdþ ¼ð Þ n Q r j¼ j. Proof As each D i is the matrix J I where J is the all ones matrix and I is the identity matrix of dimension i i, it is easy to see that detðd i Þ¼ð Þ i ð i Þ and #ðd i Þ¼ð Þ P i i (the #ðd i Þ calculation is immediate if we use [3, Lemma ]). Since r i¼ i ¼ n þ r, the equality of the theorem follows from Theorem. For a block rah, consider the jv()j-dimensional column vector defined as follows. Let a vertex v 2 V be in k cliques of sizes, 2,..., k (where each i 4 ). Let " # v ¼ Xk ðk Þ: ð2þ i¼ i For the block rah iven in Fiure, we have ¼ 63 60, and t ¼ ( /4, /4, /4, /4, 3/0, /5, /5, /5, 7/5, /3, /3). LEMMA P Let be a block rah and let be the vector defined above. Then, v2v() v ¼. Proof By induction on b, we have the number of blocks of, with the case b ¼ bein clear. When has more than one block, let H be any leaf block (i.e. a block whose deletion does not disconnect ) connected throuh cut-vertex c. Clearly, a leaf block H exists and let F ¼ {H c} be the smaller rah obtained by deletin H c from. Let H be a -clique (i.e. H ¼ K ). By induction, for the rah F, we know S ¼ P v2v(f) v ¼. It is simle to note that when we move to from F, the
vector is different from that for F only for the vertices of H. The chane in for is easily seen to be (/ ) for c and / for the other vertices of H. Thus the sum of the chanes is zero, comletin the roof. LEMMA 2 Let D be the distance matrix of a block rah. Let jvj¼n and be the vector defined by Equation (2). Let be the n-dimensional vector with all comonents equal to. Then D ¼, where is as iven in Equation (). Proof We aain induct on b, the number of blocks of with the case b ¼ bein simle. Delete a leaf block H connected to throuh c and let F ¼ {H c}. Let H be a -clique and let D F be the distance matrix of F. Let F be the vector restricted to vertices of F. Let be the column of D F corresondin to the vertex c. The v-th comonent of is v ¼ d v,c where d u,v is the distance between vertices u, v 2 F. It is simle to note that 0 D F þ F þ F ð þ F Þ t 0 D ¼ B... @... C A : ð þ F Þ t 0 If F is the restriction of to F, then by induction we have D F F ¼ F F. Here F is the vector as in Equation () for the rah F. Let t ¼ D and for v 2 F and let R v (D F ) be the v-th row of D F. For vectors a, b with identical dimension, ha, bi denotes the usual (real) inner roduct of two vectors. For a vertex v 2 F, the v-th comonent of t is t v ¼hR v ðd F Þ, F iþð Þ v þð v þ Þ. Hence, t v ¼ F þ. Thus for all vertices in F, we have ¼ F þ. For vertices u 2 H {c}, we have t u ¼hðþ F Þ t, F iþ 2 þ ¼ F þ X ð F Þ v þ v 2 F ¼ F þ, where in the first line we have used the fact that c ¼ 0 and in the second line we have used Lemma. Thus, for all vertices u 2 V(), t u ¼ F þ. Since ¼ F þ, the roof is comlete. Let have vertex set [n] and blocks i where i r. Each i is also considered as a rah on [n] with erhas isolated vertices and let its ede set be E i (i.e. i is a clique on say i vertices, but consider it as a rah on [n]). Let L i be the Lalacian of i ¼ ([n], E i ). Let I be the jvjjvj identity matrix. Define ^L ¼ Xr i¼ i L i : LEMMA 3 With the above notation, ^LD þ I ¼ t. Linear and Multilinear Alebra 395 Proof We aain induct on b, the number of blocks of with the base case b ¼ bein simle. Let H, F, c be as in the roof of Lemma and let H be a -clique. Let ^L F be the combination of the Lalacian as before, but only for the blocks of F and let D F be the distance matrix of F. Similarly, let I F be the identity matrix of order jfjjfj.
396 R.B. Baat and S. Sivasubramanian Let e c be the jfj-dimensional column vector with a in osition c and zero elsewhere and let ¼ D F e c. Let H c be a (jhj )-dimensional all ones column vector. H is a leaf-block, but considerin it as a rah on [n], let its Lalacian be denoted by L H. Let L H c be L H restricted to the set of vertices V(H) {c} and D H c the distance matrix of, restricted to the set of vertices V(H) {c}. We clearly have 0 ^L ¼ B @ ðe c t H c Þ C A, ^L F þ ðe c e t c Þ ð H c e t c Þ L H c D F ð þ F Þð H Þ t D ¼ H ðþ F Þ t D H c We need to show that for all i, j, ð ^LD þ I Þ i, j ¼ i. For rows i 2 F Z {c}: For such a row i and for columns j 2 F, we have by induction, ^L F D F þ I F ¼ð F Þ i. We denote the i-th row (and j-th column) of matrix M as R i (M) (and C j (M), resectively). Since i ¼ ( F ) i for i 2 F c, we are done for all columns in F. For columns j 2 H {c}, we note that ð ^LD þ I Þ i, j ¼hR i ð ^L F Þ, þ F i. Since the row-sum of a linear combination of Lalacians is zero, hr i ð ^L F Þ, F i¼0. Thus ð ^LD þ I Þ i, j ¼ð^LD þ I Þ i,c ¼ð F Þ i ¼ i. For rows i 2 H Z {c}: For such rows i, it is easy to see that i ¼. For all columns j 2 H, since the entries ^L i,x 6¼ 0 only if x 2 H, it is simle to see that ð ^LD þ I Þ i, j ¼.In the above result, in case i ¼ j, since the diaonal entry D i,i ¼ 0, we et a from the identity matrix to et ð ^LD þ I Þ i, j ¼ þð Þ ¼. For columns j62 H, usin the matrices L H c and D H c, we see that ð ^LD þ I Þ i, j ¼hR i ðl H c Þ, C j ðd H c Þi. Since C j ðd H c Þ¼C c ðd H c Þþd j,c H and since the column sum of a Lalacian is zero, we et ð ^LD þ I Þ i, j ¼ð^LD þ I Þ i,c ¼ ¼ i. For the row c: We need to show that for any column v 6¼ c, hr c ð ^LÞ, C v ðdþi ¼ c and that for column c, hr c ð ^LÞ, C c ðdþi þ ¼ c. We first show that hr c ð ^LÞ, C c ðdþi þ ¼ c. By induction, we know that hr c ð ^L F Þ, C c ðd F Þi þ ¼ð F Þ c. Since d c,c ¼ 0, the ð Þ required roduct is easily seen to be hr c ð ^L F Þ, C c ðd F Þi þ, which is c. We now show for v 6¼ c, hr c ð ^LÞ, C v ðdþi ¼ c. First, consider columns v 2 F {c}. By induction, we know that hr c ð ^L F Þ, C v ðd F Þi ¼ ð F Þ c. Since ^L c,c ¼ð^L F Þ c,c þ and for all u 2 H, ^L c,u ¼, we et hr cð ^LÞ, C v ðdþi ¼ ð F Þ c þ d v,c 0. Since c ¼ð F Þ c þ, we are done. Next consider columns v 2 H {c}. We have just shown that hr c ð ^LÞ, C c ðdþi ¼ c. The column vectors C v (D) and C c (D) only differ in the entries corresondin to row c and v, when restricted to rows in H and differ for all entries in F: each entry of C v (D) is larer than the corresondin entry of C c (D) by. Since a linear combination of the Lalacian has zero row-sum, we have hr c ð ^LÞ, C v ðdþi ¼ hr c ð ^LÞ, C c ðdþi þ þ, where the term arises as ^L c,c ¼ð^L F Þ c,c þ and D c,v ¼ and the term arises as ^L v,c D c,c ¼ is to be subtracted from hr cð ^LÞ, C c ðdþi. Thus, we have hr c ð ^LÞ, C v ðdþi ¼ c þ þ ¼ c, comletin the roof. THEOREM 3 Let ^L, D, and be as above. Then D ¼ ^L þ t. Proof By Lemma 3, we see that ^LD þ I ¼ t. By Lemma 2, we et t D ¼ t or t D ¼ t where clearly 6¼ 0. Thus, ^LD þ I ¼ t D. i.e. I ¼ð ^L þ t ÞD, comletin the roof. :
Linear and Multilinear Alebra 397 Theorem 3 says that even if all the blocks of are arbitrary sized cliques, D is a scalar multile of a kind of Lalacian matrix lus a constant multile of a rank one matrix. The followin known corollaries are easily derived from Theorem 3. COROLLARY [4] Let D be the distance matrix of a tree T on n vertices and let L be its Lalacian matrix. Let be the n-dimensional column vector with u ¼ 2 de u, where de u is the deree of vertex u in T. Then D ¼ L 2 þ 2ðn Þ t. COROLLARY 2 [5] Let D be the distance matrix of a rah on n vertices, all of whose blocks are K 3 s and let L be its Lalacian matrix. Let be the n-dimensional column vector with u ¼ 3 de u, where de u is the deree of vertex u in. Then D ¼ L 3 þ 3ðn Þ t. Acknowledements Some results in this work were in their conjecture form, tested usin the comuter ackae Sae. We thank the authors for enerously releasin their software as an oen-source ackae. We sincerely thank the referees for brinin several inaccuracies to our notice, resultin in a considerable imrovement in the resentation. R.B. Baat ratefully acknowledes the suort of the JC Bose Fellowshi, Deartment of Science and Technoloy, overnment of India. S. Sivasubramanian thanks Professor Murali K. Srinivasan for his suort in makin a tri to ISI Delhi ossible and the Stat-Math unit of ISI Delhi for their hositality. References [] R.B. Baat, A.K. Lal, and S. Pati, A q-analoue of the distance matrix of a tree, Linear Alebra Al. 46 (2006),. 799 84. [2] A. Brandsta dt, V.B. Le, and J.P. Sinrad, rah Classes: A Survey, SIAM Monorahs on Discrete Mathematics and Alications, SIAM, Philadelhia, PA, 999. [3] R.L. raham, A.J. Hoffman, and H. Hosoya, On the distance matrix of a directed rah, J. rah Theory (977),. 85 88. [4] R.L. raham and L. Lovasz, Distance matrix olynomials of trees, Adv. Math. 29 (978),. 60 88. [5] S. Sivasubramanian, q-analos of distance matrices of hyertrees, Linear Alebra Al. 43 (2009),. 234 248.