Ann. Funct. Anal. 5 (2014), no. 2, A nnals of F unctional A nalysis ISSN: (electronic) URL:

Ann Funct Anal 5 (2014), no 2, 127 137 A nnals of F unctional A nalysis ISSN: 2008-8752 (electronic) URL:wwwemisde/journals/AFA/ THE ROOTS AND LINKS IN A CLASS OF M-MATRICES XIAO-DONG ZHANG This paper is dedicated to Professor Tsuyoshi Ando Communicated by Q-W Wang Abstract In this paper, we discuss exiting roots of sub-kernel transient matrices P associated with a class of M matrices which are related to generalized ultrametric matrices Then the results are used to describe completely all links of the class of matrices in terms of structure of the supporting tree 1 Introduction Let I be a finite set and I = n U = (U ij, i, j I) is ultrametric matrix if it is symmetric, nonnegative and satisfies the ultrametric inequality U ij min{u ik, U kj } for all i, j, k I The ultrametric matrices have an important property that if U is nonsingular ultrametric matrix, then the inverse of U is row and column diagonally dominant M matrix (see [7] and [13]) A construction also was given in [13] to describe all such ultrametric matrices Later, nonsymmetric ultrametric matrices were independently defined by McDonald, Neumann, Schneider and Tsatsomeros in [11] and Nabben and Varge in [14], ie, nested block form(nbf) matrices and generalized ultrametric(gu) matrices After a suitable permutation, every GU matrix can be put in NBF They satisfy ultrametric inequality and are described with dyadic trees in [11], [14] and [4] On the inequality of M matrices and inverse M matrices, Ando in [1] presents many nice and excellent inequalities which play an key role in the nonnegative matrix theory Zhang [15] characterized equality cases in Fisher, Oppenheim and Ando inequalities For more detail information on inverse M and Z matrices, the reader is referred to [6], [8], Date: Received: September 23, 2013; Accepted: December 4, 2013 2010 Mathematics Subject Classification Primary 15B48; Secondary 47B99, 6010 Key words and phrases Inverse M matrix, tree, exiting root, link 127

128 X-D ZHANG [9], [10], [16] and the references in there In this paper, we follows closely the global frame work and notation on generalized ultrametric matrices supplied by Dellacherie, Martínez and Martín in [4] Recently, Nabben was motivated by the result of Fiedler in [5] and defined a new class of matrices: U-matrices (see Section 2) which satisfy ultrametric inequality and are related to GU matrices There is a common characterization in these matrices that if they are nonsingular, then their inverses are column diagonally dominant M matrices For each η η(u) = max{(u 1 ) ii, i I}, define matrix P = E η 1 U 1, where E is the identity matrix Then P is sub-markov kernel: P ij 0, for all i, j I and 1 t P 1 t (entry-wise), where 1 is the column vector of all ones Therefore ηu = (E P ) 1 = m 0 and U is proportional to the potential matrix associated to the transient kernel P Since P ij > 0 if and only if (U 1 ) ij < 0 for i j, the existence of links between different points does not depend on η, while the condition P ii > 0 depends on the value of η Define the potential vector µ = µ U associated to U by µ := U 1 1 and its total mass µ := 1 t µ Note that the following equivalence holds P m µ i > 0 (U 1 1) i > 0 (P 1) i < 1 Every i satisfying this property is called an exiting root of U(or of P ) The set of them is denoted by R := R U The potential vector ν := ν U associated to U t is given by ν := (U t ) 1 1 and ν := 1 t ν Notice that µ = ν, since 1 t ν = 1 t (U t ) 1 1 = 1 t µ Our main results in this paper are to characterize the following properties (which do not depend on η) i is a exiting root of P and P t ; and link of P, ie, for a given couple i j, whether P ij > 0 for U-matrices These properties and other related problems were completely investigated for symmetric ultrametric matrices and GU matrices in [3] and [4], respectively In Section 2, we revisit U- matrices by means of dyadic tree and give some preliminary results which are very useful In Section 3, we describe exiting root of P and P t with associated trees In Section 4, we characterize completely the links of P 2 U matrices A tree (T, ) is a finite unoriented and connected acyclic graph For (t, s) T T, t s, there is a unique path geod(t, s) of minimum length, which is called the geodic between t and s, while geod(t, t) = {t} which is of length 0 Sometime, we use geod(t, s) to stand for its edge set Fixed r T, we call it the root of tree T If s geod(t, r), we denote s t, which is a partial order relation on T For s, t T, s t = sup{v, v geod(s, r) geod(t, r)} denotes the closest common ancestor of s and t The set of successors of t is s(t) = {s T, s t, (s, t) } Then I( ) = {i T, s(i) = } is the set of leaves of the tree T A tree is said dyadic if s(t) = 2 for t / I( ) The successors of t are denoted by t and t + For t T, the set L(t) := {i I( ), t geod(i, r)} completely characterizes t Hence we can identify t and L(t) In particular, r is identified with L(r) = I( ) and i I( ) with the singleton i Hence we can assume that each vertex of T

THE ROOTS AND LINKS IN A CLASS OF M-MATRICES 129 is a subset of the set of leaves I( ) The distinction between the roles of L, as L T (mean that L is a vertex of tree T ) and L I (mean that L is regarded as the subset of I( ) corresponding to the vertex of T ), will be clear in the context when we use them By the above notations and definition of GU matrices in [4], The definition of U matrices in [12] may be restated in the following way Definition 21 U = (U ij : i, j I) is a U matrix if there exists a dyadic tree (T, ) with fixed a root r and a leaf n I, and nonnegative real vectors α = (αt : t T ), β = (β t : t T ) satisfying (i) I = I( ), α I = β I ; and α t = α t n for t r +, t / I; (ii) α t β t for t T ; (iii) α and β are - increasing, ie, t s implies α t α s and β t β s ; (iv) t + geod(r, n) for t geod(r, n) and t n; and α t = β t for t geod(r, n) (v) U ij = α t if (i, j) (t, t + ) and U ij = β s if (i, j) (t +, t ), where t = i j and s = max{i j, i n}; U ii = α i = β i for i I We say that (T, ) support U and U is U associated with tree (T, ) It is easy to show that this definition is equivalent to Definition 21 in [12] Observe that for each L T, the matrix U L L is either GU or U matrix, where the GU matrix consistent with the definition of GU matrix in [4] The tree supporting it, denoted by (T L, L ), is the restriction of (T, ) on L and the associated vectors which are the restrictions of α and β on T L In other words, (T L, L ) is the subtree of (T, ) with the root L and the leaves set L The potential vectors and the exiting roots of U L, U t L are denoted, respectively by µ L, ν L, R L, R t L The sub-kernel corresponding to U L, U t L is denoted by P L, (P t ) L If U is nonsingular U matrix, it can be shown that U L is also nonsingular GU or U matrix by Schur decomposition and induction argument We now introduce the following relation in the set of leaves I For i j, we say i < j if i t, j t + with t = i j Assume that I = {1, 2,, n} By permuting I, we can suppose is the usual relation on I Therefore, we will assume that this is standard presentation of U matrices in this paper In the other words, Let U U and I = I I + Denote := I and K := I + Thus ( ) U α U = I 1 1 t K b K 1 t, U K where α I = min{u ij : i, j I} and b K = U K e K with e K = (0,, 0, 1) t unit vector, ie, b K is the last column of U K Note that in here, U is GU matrix and U K is still U matrix also, which has a similar 2 2 block structure, and its the first diagonal block is a special GU matrix We begin with the following theorem in which we re-prove some known result in [12] Theorem 22 [12] If U is nonsingular U matrix, then (i) α I µ < 1 and ( ) C D U 1 =, E F where

130 X-D ZHANG (ii) C = E = U 1 + α I 1 α I µ µ ν t, D = α I µ ν t K, µ I = 1 1 α I e µ K ν t, ( 1 α I µ K 1 α I µ µ µ K µ (1 α Iµ K ) 1 α I µ F = U 1 K + α Iµ e K ν t K e K ) ( 0 ; ν I = νk (iii) µ I = µ K (iv) (µ I ) i 0 for i = 1, 2,, n 1 (v) (ν I ) i = 0 for i = 1, 2,, n 1; and (ν I ) n = µ I = 1 Proof Since U is nonsingular, U is nonsingular GU matrix By Theorem 36(i) in [11], α µ 1, where α is smallest entry in U Hence α I µ α µ 1 Suppose that α I µ = 1, by theorem 36(ii) in [11], U has a row whose entries are all equal to α I Noting that the last row whose entries are equal to, there are two rows which are proportional, which implies U is singular, a contradiction Therefore α I µ < 1 By Schur decomposition and the inverse of matrix formula, it is not difficult to show that the rest of (i) holds Since µ = ν and µ K = ν K, C1 + D1 K = U 1 1 α I + µ ν 1 α I µ 1 t + α I µ ν 1 α I µ K1 t K E1 + F 1 K = = 1 α Iµ K µ, ) 1 e K ν 1 α I µ 1 t + U 1 K 1 K + α Iµ e K ν 1 α I µ K1 t K = µ K µ (1 α I µ K ) e K, 1 t C + 1 t KE = 1 t U 1 + = 0, 1 t D + 1 t KF = α I 1 t 1 α I µ µ νk t + 1 t KU 1 K = νk t So (ii) holds Furthermore, α I 1 t 1 α I µ µ ν t + 1 t 1 K e K ν t + 1t K µ I = 1 α Iµ K µ + µ K µ (1 α I µ K ) = µ K α I µ e K νk t 1 Thus (iii) holds Since e t I U = 1t, ν I = 1 e I which implies µ I = ν I = 1 By (iii), we have 1 α I µ K = 1 α I µ I 1 α I 0 Hence it is easy to show that (iv) and (v) hold by using the induction on the dimension of U

THE ROOTS AND LINKS IN A CLASS OF M-MATRICES 131 3 Exiting roots of P In order to characterize the exiting roots of P, we introduce some notations and symbols Let U be a U matrix with supporting tree (T, ) and fixed a root r and a leaf n For i I( ), denote by N + i = {L T : L i, α L = α i } and N i = {L T : L i, β L = β i } Now we can construct the set Γ t : for L / geod(r, n), (L, L ) Γ t if and only if there exists a i L + such that L N i ; (L, L+ ) Γ t if and only if there exists a i L such that L N + i For L geod(r, n), (L, L ) Γ t and (L, L + ) / Γ t Theorem 31 Let U be nonsingular U Then (i) R t I = {n} (ii) For L T, i R t L if and only if geod(i, L) Γt = Proof (i) follows from Theorem 22 (v) We prove the assertion (ii) by using an induction on the dimension n of U It is trivial for n = 1, 2 Assume that ( ) U α U = I 1 1 t K b K 1 t U K If L I, then U L = (U ) L is GU matrix By Theorem 3 in [4] and (r, r ) / geod(i, L), the assertion (ii) holds If L I +, then U L = (U K ) L is still U matrix By the hypothesis and (r, r + ) / geod(i, L), i R t L if and only if geod(i, L) Γt = If L = I, then for i n, (i n, (i n) ) Γ t and geod(i, L) Γ t ; for i = n, geod(i, L) Γ t = Hence by (i), L T, i R t L if and only if geod(i, L) Γ t = In order to describe the exiting of P, we construct the set Γ: For each L T, (L, L ) Γ, (L, L + ) Γ, if and only if there exists i L +, i L, such that L N + i, L N, respectively Theorem 32 Let U = (U ij, i, j I) be a nonsingular U matrix Then (i) n i R if and only if geod(i, I) Γ = (ii) If L T, then n i R L if and only if geod(i, I) Γ = Proof (i) We prove the assertion by the induction It is trivial for n = 1, 2 We assume that ( ) U α U = I 1 1 t K b K 1 t, U K where α I = min{u ij : i, j I}, b K = U K e K with e K = (0,, 0, 1) t unit vector, U is GU matrix and U K is U matrix By Theorem 22(ii), we have ( ) µ I = 1 α I µ K 1 α I µ µ µ K µ (1 α Iµ K ) 1 α I µ Now we consider the following two cases Case 1: i I Then i R if and only if i R and 1 α I µ K > 0 by Theorem 22 (ii) Note that µ I = µ K from Theorem 22 (iii) 1 α I µ K = 0 if and only if α I = α n if and only if (I, I ) Γ because it follows from the definition of Γ, and if there exists an n q I + such that I N q + which implies α I = α q by definition of 21(i) Therefore each entries of the q th row is α q which implies e K

132 X-D ZHANG that U is singular Hence 1 α I µ K > 0 if and only if (I, I ) / Γ By the induction hypothesis, n i R if and only if geod(i, I) Γ = Case 2: i I + Suppose that (I, I + ) Γ Then there exists j 0 I such that I N j 0 which implies β I = β j0 Hence β j0 = α I follows from α I = β I So U is singular, a contradiction Therefore (I, I + ) / Γ By Theorem 22 (ii), n i R if and only if n i R K Because U K is U matrix, n i R if and only if geod(i, I + ) Γ = by the induction hypothesis, so if and only geod(i, I) Γ = (ii) Since U L is GU matrix or U matrix, the assertion follows from Theorem 3 in [4] or (i) Theorem 33 Let U be a U matrix Then (i) U 1 is row diagonal dominant M matrix if and only if µ µ n L (1 α L µ L +) (31) 1 α L µ L L geod(r,n),l n (ii) n R if and only if (31) becomes strict inequality Proof From Theorem 22 (ii), the sum of n th row of U 1 is (µ K ) n µ I (1 α Iµ I +), 1 α I µ I where (µ K ) n is the last component of µ K Hence by the induction hypothesis, the sum of n th row of U 1 is µ µ n L (1 α L µ L +) 1 α L µ L L geod(r,n),l n Therefore (i) follows from Theorem 22 (iii) (ii) is just a consequence of (i) and the definition of exiting Lemma 34 Let U = (U ij, i, j I) a nonsingular GU matrix Then U ii µ I 1 for all i I Moreover, max{u ii, i I}µ I = 1 if and only if there exist a column whose entries are equal to max{u ii, i I} Proof Since U is a GU matrix and 1 = UU 1 1 = Uµ, 1 = n j=1 U ij(µ I ) j n j=1 U ii(µ I ) j Hence we have U ii µ I 1 for i I Let max{u ii } = U qq and suppose that U qq µ I = 1 Then 1 = n j=1 U ij(µ I ) j n j=1 U ii(µ I ) j U qq µ I = 1 Hence n j=1 (U ij U qq )µ j = 0 for i I, which yields the result Conversely, let 1 max{u ii } = U qq and e q = (0,, U qq,, 0) t Then Ue q = 1 Hence µ = e q and U qq µ I = 1 Theorem 35 Let U be a nonsingular U matrix If there exists i with i n such that U ii < Then U is not a row diagonally dominant matrix, neither n is a root of P Proof We prove the assertion by the induction on the dimension of matrix U It is easy to show that the assertion holds for order n = 1, 2 Now we assume that ( ) U α U = I 1 1 t K b K 1 t U K

THE ROOTS AND LINKS IN A CLASS OF M-MATRICES 133 If there exists a i K such that (U) ii <, then by the induction hypothesis, the last component of µ K is less than 0 So (µ I ) n < 0 by theorem 22(ii) Hence we assume that there exists a i such that U ii < Then by Lemma 34, µ 1 (U ) ii = 1 U ii Hence i K (µ I ) i = µ K µ (1 α I µ K ) = µ K µ 1 ( 1 1 ) < 0, U ii since µ K = µ I = 1 by Theorem 22 (iii) and (v) On the other hand, (µ I ) j 0 for i K \ {n} Therefore, (µ I ) n < 0 So the assertion holds Corollary 36 Let U be a nonsingular U matrix If there exists a i with i n such that U ii then n is not root of P Proof It follows from Theorem 35 and its proof Lemma 37 Let U be a U matrix If j I U nj j I U ij for i = 1, 2,, n 1 Then U is a row and column diagonally dominant M-matrix and n is an exiting of P Proof From U 1 U = I n, 1 = j I l I (U 1 ) nl U lj = l I, l n j I U lj(u 1 ) nl + j I U nj(u 1 ) nn Since (U 1 ) nl 0 for l n and j I U nj j I U lj for l n, 1 l I, l n j I U nj(u 1 ) nl + j I U nj(u 1 ) nn = j I U nj l I (U 1 ) nl Hence l I (U 1 1 ) nl P > 0 Hence the result follows form theorem j I U nj 22 4 Links of P In this section, we describe completely the links of transient kernel P associated with a class of U matrices Firstly we give some lemmas Lemma 41 Let U be a nonsingular U matrix Then for i, j I =, i j, (U 1 ) ij < 0 if and only if (U 1 ) ij < 0 and U ij > α I Proof Necessity By Theorem 22, we have (U 1 ) = U 1 α I + µ ν t = (U (α I )) 1, where U (α I ) = U α I 1 1 t is a nonsingular GU matrix Hence for i, j I =, (U 1 ) ij < 0 implies (U 1 ) ij < 0 Further, by Theorem 36 in [14], (U 1 ) ij = (U (α I )) 1 ij < 0 implies that (U (α I )) ij > 0 So U ij > α I Sufficiency Note that α I < min{(u ) ii } (otherwise every entries of some row rather than n is α I which yields U is singular) Note that U is a GU matrix By Theorem 6 in [4], (U 1 ) ij < 0 implies that (U 1 ) ij = (U (α I ) 1 ) ij < 0 Lemma 42 Let U be a nonsingular U matrix of order n Then (i) for i I =, j I + = K, (U 1 ) ij < 0 if and only if i R and j = n (ii) for i I +, j I, (U 1 ) ij < 0 if and only if i = n and j R

134 X-D ZHANG Proof (i) For i I =, j I + = K, by Theorem 22 (i), (U 1 ) ij = ( α I 1 α I µ µ ν t K ) ij < 0 if and only if (µ ) i > 0 and (ν K ) j > 0 if and only if i R and j = n (ii) For i I =, j I + = K, by Theorem 22, (U 1 ) ij < 0 if and only if 1 ( 1 α I e µ K ν t ) ij < 0 if and only if i = n and j R Lemma 43 Let U be a nonsingular U matrix Then for i I + = K, j I + and i j, (U 1 ) ij < 0 if and only if (U 1 K ) ij < 0 Proof The assertion follows from Theorem 22 (V) Now we can state the main result in this section Theorem 44 Let U be a nonsingular U matrix associated supporting tree (T, T ) with fixed the leaf n Suppose that i j and i j = L (i) If L geod(i, n), then P ij > 0 if and only if (P L ) ij > 0; ie, if and only if (ia) for i L, j L +, i R L and j = n (ib) for i I +, j I, i = n and j R L (ii) If L / geod(i, n) and L 1 = (i j) n, then (iia) (P L ) ij > 0 if and only if i R L and j R L + for i L, j L + ; and i R L + and j R L for i L +, j L (iib) (P L 1 )ij > 0 if and only if (P L ) ij > 0; and either U ij > α L, or U ij = α ij and for every M L such that α M = α L implies that (M, M ) / Γ t for {i, j} M, (M, M + ) / Γ for {i, j} M + hold (iic) P ij > 0 if and only if (P L 1 )ij > 0 and U ij > α L1 Proof We prove the assertion by the dimension of the matrix U It is trivial for n = 1, 2 Now assume that i j and i j = L Case 1: L geod(i, n) If L = I, then i I (or I + ) and j I + ( or I ) Hence by Lemma 42 and (2), the assertion (i) holds If L I, then L I + and i, j I + So by Lemma 43, we have P ij > 0 if and only if (U 1 ) ij < 0 if and only if (U 1 I ) + ij < 0 Since U I + is a U matrix, by the induction hypothesis, we have (U 1 ) ij < 0 if and only if (P L ) ij > 0 Moreover, the rest of (i) follows from Lemma 42 Case 2: L / geod(i, n) If L 1 = I, then i, j L I and U is a GU matrix Hence (iia) and (iib) follow from Theorem 4 in [4] At the same time, (iic) follows from Lemma 41 and (2) that P ij > 0 if and only if (U 1 I ) ij < 0 and U ij > α L1 If L 1 I, then P ij > 0 if and only if (U 1 ) ij < 0 if and only if (U 1 I ) + ij < 0 if and only if (P I+ ) ij > 0 By the induction hypothesis and Theorem 4 in [4], the assertions of (i) and (ii) hold Corollary 45 Let U be a nonsingular U matrix A B n, then (U 1 ) ij = 0 for i A and j B If A, B geod(i, n) and Proof Since A B = A or B, The result follows from Theorem 44 (i)

THE ROOTS AND LINKS IN A CLASS OF M-MATRICES 135 Corollary 46 Let U be nonsingular U matrix with supporting tree If β i = β t for all t i and all i I, then the inverse of U has the following structure: W 11 0 0 W 1s U 1 = 0 W 22 0 W 2s, W s1 W s2 W s,s 1 W ss where W ii is a lower triangular matrix for i = 2, 3, s 1; W ss is a 1 1 matrix Moreover, if β i > β t for all t i and all i I; and both α I > α I and β A > β A for A geod(i, n); then each entry of W 11 is nonzero, each entry of W is and W si is nonzero for i = 1,, s; and each each entry of the lower part of lower triangular matrix W ii is zero for i = 2,, s 1 Proof We partition the blocks of U 1 = (W ij ) corresponding to the leaves sets of vertices of geod(i, n) In particular, W ss is corresponding to fixed vertex n By Corollary 45, W ij = 0 for i j s Further, it follows from Theorem 44(iic) that W ii is a lower triangular matrix for i = 2,, s 1, since for A I +, α A = α A n Let n A geod(a, n) Since β i > β t for all t i and all i I, U A is a strictly generalized ultrametric matrix Hence by Theorem 44(iia) and (iib) or Theorem 35 in [14], each entry of W 11 is nonzero, since α I > α I ; and each entry of the lower part of the lower triangular matrix W ii is nonzero for i = 2,, s 1, since β A > β A = α A Moreover, since R A = A, each entry of W is and W si is nonzero by Theorem 44 (i) The proof is finished Remark 47 From Theorem 44, it is easy to see that the links of U U are not involved in whether n R or not Hence we may directly determine whether each entries of U 1 is zero or not from the structure of support tree Let us give an example to illustrate Theorems 32 31, 44 Example 48 Let U be a U matrix of order 7 with support tree (T, ) as in the Figure 1, where I is root and 6 is fixed leaf 1(3, 3) 2(3, 3) 3(4, 4) 4(4, 4) 5(4, 4) 6(4, 4) + + + C(2, 4) D(3, 3) A(2, 3) + B(2, 2) + I(1, 1) Fig 1 Then the matrix U and inverse of U are

136 X-D ZHANG U = 3 2 1 1 1 1 3 3 1 1 1 1 2 2 4 2 2 2 2 2 4 4 2 2 3 3 3 3 4 3 4 4 4 4 4 4 U 1 = 1 8 It is easy to see that Γ = {(A, 2), (C, 4)} and 8 4 0 0 0 1 8 8 0 0 0 0 0 0 4 0 0 2 0 0 4 4 0 0 0 0 0 0 8 6 0 4 0 4 8 11 Γ t = {(A, 1), (C, 3), (I, A), (B, C), (D, 5)} By Theorem 32 and Corollary 33, we have R = {1, 3, 5} Further, we determine all links of P by Theorem 44 For instance, in order to determine P 43, we consider 3 4 = C / geod(i, 6) and (3 4) 6 = B By Theorems 32 and 31, 4 R C, 3 R t C + Hence by Theorem 44 (iia), (P C ) 43 > 0 Further, by Theorem 44(iib) and U 43 = 4 > α C = 2, (P B ) 43 > 0 Therefore P 43 > 0 follows from Theorem 44(iic) and U 43 = 4 > α B = 2 Acknowledgement The research is supported by National Natural Science Foundation of China (No11271256), Innovation Program of Shanghai Municipal Education Commission (No14ZZ016)and Specialized Research Fund for the Doctoral Program of Higher Education (No20130073110075) The author would like to thank anonymous referees for their comments and suggestions References 1 T Ando, Inequalities for M -matrices, Linear Multilinear Algebra 8 (1979/80), no 4, 291 316 2 A Berman and RS Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic 1979, SIAM 1994, New York 3 C Dellacherie, S Martínez and S Martín, Ultrametric matrices and induced Markov chains, Adv in Appl Math 17 (1996), 169 183 4 C Dellacherie, S Martínez and S Martín, Description of the sub-markov kernel associated to generalized ultrametric matrices: an algorithmic approach, Linear Algebra Appl 318 (2000), 1 21 5 M Fiedler, Special ultrametric matrices and graphs, SIAM Matrix Anal Appl 22 (2000), 106 113 6 CR ohnson and RL Smith, Inverse M -matrices, II, Linear Algebra Appl 435 (2011), no 5, 953 983 7 S Martínez, G Michon and S Martín, Inverses of ultrametric matrices are of Stieltjes types, SIAM Matrix Anal Appl 15 (1994), 98 106 8 S Martínez, S Martín and X-D Zhang,A new class of inverse M-matrices of tree-like type, SIAM Matrix Anal Appl 24 (2003), no 4, 1136 1148 9 S Martínez, S Martín and X-D Zhang, A class of M matrices whose graphs are trees, Linear Multilinear Algebra 52 (2004), no 5, 303 319 10 McDonald, R Nabben, M Neumann, H Schneider and M Tsatsomeros, Inverse tridiagonal Z matrices, Linear Multilinear Algebra 45 (1998), 75 97 11 McDonald, M Neumann, H Schneider and M Tsatsomeros, Inverse M-matrix inequalities and generalized ultrametric matrices, Linear Algebra Appl 220 (1995), 329 349 12 R Nabben, A class of inverse M-matrices, Electron Linear Algebra 7 (2000), 353 358

THE ROOTS AND LINKS IN A CLASS OF M-MATRICES 137 13 R Nabben and RS Varga, A linear algebra proof that the inverses of strictly ultrametric matrix is a strictly diagonally dominant are of Stieljes types, SIAM Matrix Anal Appl 15 (1994), 107 113 14 R Nabben and RS Varga, Generalized ultrametric matrices- a class of inverse M- matrices, Linear Algebra Appl 220 (1995), 365 390 15 X-D Zhang, The equality cases for the inequalities of Fischer, Oppenheim, and Ando for general M-matrices, SIAM Matrix Anal Appl 25 (2003), no 3, 752 765 16 X-D Zhang, A note on ultrametric matrices, Czechoslovak Math 54(129) (2004), no 4, 929 940 Department of Mathematics and MOE-LSC, Shanghai iao Tong University, Shanghai 200240, PRChina E-mail address: xiaodong@sjtueducn