Four-point condition matrices of edge-weighted trees

Ali Azimi; Rakesh Jana; Mukesh K. Nagar; Sivaramakrishnan Sivasubramanian

doi:10.1515/spma-2024-0011

Artikel Open Access

Four-point condition matrices of edge-weighted trees

Ali Azimi , Rakesh Jana , Mukesh K. Nagar und Sivaramakrishnan Sivasubramanian

Veröffentlicht/Copyright: 2. Juli 2024

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Abstract

Formulas for the determinant of distance matrix D T of tree T are known in the unweighted case and in the case when the edges of T have commuting variable weights. Associated with the four-point condition (4PC) and a tree T are two matrices, the Max4PC T and the Min4PC T . These are not full rank matrices and their rank, a basis B , and formulas for the determinant when restricted to the rows and columns of B are known. In this work, we generalize both these matrices to the case when the edges of T have commuting variable weights and determine edge-weighted counterparts of known results.

Keywords: distance matrix; tree; four-point condition; edge weights

MSC 2010: 05C05; 05C12; Secondary 15A15

1 Introduction

Let T = ( V , E ) be a tree on n vertices with edge set E = { e 1 , … , e n − 1 } . For 1 ≤ i ≤ n − 1 , assign a variable α i on the edge e i . We assume that the α i ’s commute with each other and think of α i as the weight on edge e i . For two vertices i , j ∈ T , define their weighted distance w i , j as the sum of the weights of the variables on the edges in the unique path between i and j . We will need the unweighted distance d i , j between vertices i and j as well in this work. Recall that d i , j is the number of edges on the path between i and j . The weighted distance matrix WD of T is an n × n matrix whose ( i , j ) th entry is the weighted distance w i , j . Clearly, when we set α i = 1 for 1 ≤ i ≤ n − 1 , the matrix WD reduces to the unweighted distance matrix D = ( d i , j ) of T . The following is a remarkable result of Graham and Pollak on the distance matrix D of a tree T [15].

Theorem 1

(Graham and Pollak) Let T be a tree on n vertices with unweighted distance matrix D. Then, det D = ( − 1 ) n − 1 2 n − 2 ( n − 1 ) .

Thus, the value of det D is independent of the structure of the tree T and only depends on n , the number of vertices of T . Bapat et al. [6] considered the weighted distance matrix WD of a tree T on n vertices with edge e i ∈ E ( T ) having weight α i for 1 ≤ i ≤ n − 1 . The following two quantities appear multiple times in this work, and so for a tree T on n vertices, we define Prod T ( α ) = ∏ i = 1 n − 1 α i and Sum T ( α ) = ∑ i = 1 n − 1 α i . With this notation, their result is the following.

Theorem 2

(Bapat, Kirkland, and Neumann) Let T be a tree on n vertices. For 1 ≤ i ≤ n − 1 , let edge e i ∈ E ( T ) have weight α i , and let WD denote T ’s weighted distance matrix. Then,

det WD = ( − 1 ) n − 1 2 n − 2 ⋅ Prod T ( α ) ⋅ Sum T ( α ) .

Thus, the value of det WD is only a function of n and the edge weights (the α i ’s), and does not depend on the structure of T . Clearly, setting α i = 1 when 1 ≤ i ≤ n − 1 in Theorem 2 gives us Theorem 1, and hence, Theorem 2 generalizes Theorem 1. These two results have inspired further studies on distances in trees (see, for instance, [5,7–12,17]).

Given a distance matrix D , a natural question is the following. When will entries of D arise from shortest path distances on a tree? Answering this, Buneman [14] showed this happens if and only if the distances satisfy the four-point condition (4PC). The 4PC states that for any four vertices i , j , k , and ℓ in T , the maximum value of the three terms d i , j + d k , ℓ , d i , k + d j , ℓ , and d i , ℓ + d j , k equals the second maximum value. Buneman mentioned in his study that the 4PC also holds in the weighted case when the edges have weights.

Let V = { 1 , 2 , … , n } be the set of vertices of T , and let V n be the set of size-2 subsets of V . Three unweighted matrices related to the 4PC having rows and columns indexed by V n are known in the literature. These are the Max4PC T matrix, the Min4PC T matrix, and the 2-Steiner distance matrix D 2 ( T ) of T .

We describe the three matrices and give known results about them. Let X , Y ∈ V n with X = { i , j } and Y = { k , ℓ } . The entry in row X and column Y in the maximum-4PC (respectively, minimum-4PC) matrix is denoted as Max4PC ( X , Y ) (respectively, Min4PC ( X , Y ) ) and is defined to be the maximum (respectively, minimum) value among three terms d i , j + d k , ℓ , d i , k + d j , ℓ , and d i , ℓ + d j , k . The Steiner distance between X and Y is denoted by ST ( X , Y ) and is defined as the number of edges in a minimum size subtree of T that contains the vertices in X ∪ Y . The entry of the 2-steiner distance matrix D 2 ( T ) in the row indexed by X and the column indexed by Y is ST ( X , Y ) . Azimi and Sivasubramanian [3] showed that as matrices,

D 2 ( T ) = 1 2 ( Max4PC T + Min4PC T ) .

Thus, for all X , Y ∈ V n , we have

(1) ST ( X , Y ) = 1 2 ( Max4PC ( X , Y ) + Min4PC ( X , Y ) ) .

Bapat and Sivasubramanian [13] studied the Min4PC T matrix for a tree T . The authors determined the rank of Min4PC T , implicitly gave a basis B for its row space, and also determined its Smith normal form (SNF). Azimi et al. [2] determined the inverse of the submatrix of Min4PC T when restricted to rows and columns indexed by this basis B . The Min4PC T matrix is a hollow matrix as studied in Bapat and Kurata [18], and hence, it would be interesting to see if some of our results hold in the hollow matrix setting.

Azimi et al. [4] studied the matrix D 2 ( C T ) of a caterpillar tree C T and among other results, determined the rank of D 2 ( C T ) . They asked for generalizing their results to all trees. Azimi and Sivasubramanian [3] considered D 2 ( T ) for all trees T . The authors determined the rank of D 2 ( T ) and gave a basis for its row space, determined the inertia, the determinant, and the inverse of the submatrix of D 2 ( T ) when restricted to rows and columns from their basis.

The matrix Max4PC T for a tree T has been recently studied by Azimi et al. [1]. The authors gave a formula for the rank of Max4PC T , gave an algorithm that outputs a class B of bases for the row space of Max4PC T , and gave a formula for det Max4PC T [ B , B ] for any basis B ∈ B . The authors also determined the inertia of Max4PC T .

In this article, we consider edge-weighted versions of two of these matrices. Let T be a tree on n vertices with edge set E = { e 1 , e 2 , … , e n − 1 } , and for 1 ≤ i ≤ n − 1 , let the edge e i have variable weight α i . As remarked earlier, we assume that the α i ’s commute with each other. For U , V ∈ V n , since the matrix entry indexed by ( U , V ) will be a polynomial with several variables (depending on both the tree T and the matrix considered), it is easy to see that setting α i = 1 for 1 ≤ i ≤ n − 1 would give us the numerical matrices described earlier. However, we take care not to set negative values to any α i as distances could then become negative and cease being a metric.

We consider edge-weighted versions of the matrices Max4PC T and Min4PC T and denote them as WtMax4PC T and WtMin4PC T , respectively. In this multi-variable setting, results on rank follow without major difficulties, but a formula for the determinant obtains interesting and proofs tend to become involved. Hence, our main results in this work are similar to how Theorem 2 generalizes Theorem 1. Among other results, we mention the following two results on the WtMin4PC T and the WtMax4PC T matrices.

Theorem 3

Let T be a weighted tree on n > 2 vertices with edge set E. For 1 ≤ k ≤ n − 1 , let edge e k ∈ E have weight α k . Take e = { i , j } ∉ E and let B = E ∪ { e } . Then, B is a basis for the row space of WtMin4PC T . Furthermore,

det WtMin4PC T [ B , B ] = ( − 1 ) n − 1 2 n − 2 ( d i , j − 1 ) 2 ⋅ Prod T ( α ) ⋅ Sum T ( α ) .

In Theorem 3, when e = { i , j } ∉ E is chosen with d i , j = 2 and B = E ∪ { e } , it is easy to see that det WtMin4PC T [ B , B ] is identical to the expression given in Theorem 2. Hence, in this case, the two weighted matrices WtMin4PC T [ B , B ] and WD have the same determinant value. We give an unweighted version in Theorem 6. It is simple to check that setting α i = 1 for 1 ≤ i ≤ n − 1 in Theorem 3 gives Theorem 6. Theorem 3 is proved in Section 2.

We move on to the WtMax4PC T matrix. As mentioned earlier, in [1, Theorem 20], an algorithm for constructing a family B of bases was given. By part (a) of Theorem 24, any output B of this algorithm contains a unique pendant edge e ∈ B . The weight of this edge e plays a role in our formula for the weighted case. In Section 4, we prove the following.

Theorem 4

Let T be a tree on n ≥ 3 vertices with p leaves. Let E = { e 1 , e 2 , … , e n − 1 } be its edge set, and for 1 ≤ i ≤ n − 1 , let α i be the weight of edge e i . Let B be a basis for the row space of WtMax4PC T output by [1, Lemma 9]. Let E B denote the set of non-pendant edges of T that are in B, and let e be the unique pendant edge in B. Let the weight of e be α . Then,

det WtMax4PC T [ B , B ] = ( − 1 ) n − p 2 2 ( n − p − 1 ) α 2 ∏ e r ∈ E B α r 2 .

Thus, for any pendant edge e i ∉ B , setting α i to any positive real value, setting α = 1 , and for any non-pendant edge e j ∈ B , setting α j = 1 , we recover our earlier result [1, Theorem 2] (this result is also quoted as Theorem 20 in this work).

2 Rank and basis of WtMin4PC T

In this section, we consider the WtMin4PC T matrix for a tree T on n vertices. Bapat and Sivasubramanian [13] showed for a tree T that the rank of its unweighted Min4PC T matrix is n . Recall that the rank of its usual distance matrix D is also n . Furthermore, the authors showed that both the matrices D and Min4PC T have the same non-zero SNF values.

We first present a class B of bases for the row space of both Min4PC T and WtMin4PC T . Additionally, we give a formula for the determinant of the submatrix of both Min4PC T and WtMin4PC T when we restrict these matrices to the indices in rows and columns from a basis B ∈ B .

Bapat and Sivasubramanian [13] showed that the determinant of Min4PC T [ B , B ] obtained by restricting Min4PC T to the rows and columns contained in B = { e 1 , … , e n − 1 , g } , where g ∉ E ( T ) divides 2 n − 2 ( n − 1 ) . We first ask when such a set B will be a basis for the row space of Min4PC T ? We answer this by providing a formula for det Min4PC T [ B , B ] .

Theorem 5

(Bapat and Sivasubramanian [13, Lemma 3, Corollary 7, Theorem 9]) Let T be an unweighted tree on n vertices and E = { e 1 , … , e n − 1 } be its edge set. Then, rank ( Min4PC T ) = n and Min4PC T [ E , E ] = 2 ( J − I ) . Let the rows and columns of Min4PC T [ E , E ] be indexed sequentially by e 1 , e 2 , … , e n − 1 . For f = { i , j } ∉ E , define a vector v f ∈ R n − 1 with its coordinates indexed by E as v f t = Min4PC T ( f , e 1 ) , … , Min4PC T ( f , e n − 1 ) . Then, we have

v f t Min4PC T [ E , E ] − 1 v f = ( n − 1 ) ( d i , j − 1 ) 2 2 ( n − 2 ) .

Our next result describes a class B of bases for the row space of Min4PC T for a tree T . We also give the determinant of the submatrix of Min4PC T when its rows and columns are restricted to elements of B ∈ B .

Theorem 6

Let T be a tree on n vertices and E = { e 1 , … , e n − 1 } be the edge set. Take f = { i , j } ∉ E and let B = E ∪ { f } . Let B denote the collection of such B over all choices of { i , j } . Then, any B ∈ B is a basis for the row space of Min4PC T . Furthermore,

det Min4PC T [ B , B ] = ( − 1 ) n − 1 2 n − 2 ( n − 1 ) ( d i , j − 1 ) 2 .

Proof

It is easy to see that Min4PC T [ B , B ] can be written in block form as follows:

(2) Min4PC T [ B , B ] = Min4PC T [ E , E ] v f v f t 0 .

Since Min4PC T [ E , E ] is invertible, by the Schur complement formula for determinants (see Horn and Johnson [16]), we obtain

det Min4PC T [ B , B ] = − ( v f t Min4PC T [ E , E ] ( − 1 ) v f ) det ( Min4PC T [ E , E ] ) = − ( n − 1 ) ( d i , j − 1 ) 2 2 ( n − 2 ) ⋅ ( − 1 ) n − 2 2 n − 1 ( n − 2 ) = ( − 1 ) n − 1 2 n − 2 ( n − 1 ) ( d i , j − 1 ) 2 .

This completes the proof.□

Remark 7

If we choose f = { i , j } in Theorem 6 with d i , j = 2 , we obtain det Min4PC T [ B , B ] = det D , where D is the usual distance matrix of T .

We generalize Theorem 6 to edge-weighted trees. Towards this, for a tree T on n vertices with edge weights α i , define a function Γ ( T ) as follows:

(3) Γ ( T ) = − ( n − 2 ) ( n − 5 ) Prod T ( α ) + ∑ i = 1 n − 1 α i 2 ∑ j = 1 , j ≠ i n − 1 ∏ k = 1 , k ≠ i , j n − 1 α k .

Lemma 8

Let T be a tree on n vertices. Let E = { e 1 , … , e n − 1 } be its edge set, and for 1 ≤ i ≤ n − 1 , let edge e i have weight α i . Let WK = WtMin4PC T [ E , E ] . Then, for e i ∈ E , we have WK ( e i , e i ) = 0 and for e j ∈ E with i ≠ j , we have WK ( e i , e j ) = α i + α j . Furthermore,

det WK = ( − 1 ) n − 2 2 n − 3 Γ ( T ) .

Proof

Let T be a tree on n vertices with edge set E = { e 1 , … , e n − 1 } and let WK = WtMin4PC T [ E , E ] . For an ( n − 1 ) × ( n − 1 ) matrix M and for 1 ≤ i ≤ n − 1 , denote the i th column of M as C i ( M ) . Note that WK can be expressed as a sum of two ( n − 1 ) × ( n − 1 ) matrices, i.e., we can write WK = A + B , where

A = α 1 α 2 … α n − 1 α 1 α 2 … α n − 1 ⋮ ⋮ ⋱ ⋮ α 1 α 2 … α n − 1 and B = − α 1 α 1 … α 1 α 2 − α 2 … α 2 ⋮ ⋮ ⋱ ⋮ α n − 1 α n − 1 … α n − 1 .

For S ⊆ [ n − 1 ] , let X S be an ( n − 1 ) × ( n − 1 ) matrix defined by C i ( X S ) = C i ( A ) if i ∈ S and C i ( X S ) = C i ( B ) if i ∉ S . Note, by the multilinearity of the determinant, that

(4) det WK = ∑ S ⊆ [ n − 1 ] det X S .

If ∣ S ∣ > 1 , then det X S = 0 , as α j C i ( X S ) = α i C j ( X S ) for some i ≠ j . Thus, in (4), we only need to consider those S ⊆ [ n − 1 ] with ∣ S ∣ ≤ 1 . Let J denote the ( n − 1 ) × ( n − 1 ) all ones matrix, and let I denote the ( n − 1 ) × ( n − 1 ) identity matrix. It is easy to see that

(5) det X ∅ = det B = Prod T ( α ) det ( J − 2 I ) = ( − 1 ) n − 2 2 n − 2 ( n − 3 ) Prod T ( α ) .

If S = { i } , note that C i ( X S ) = α i 1 . Let Cofactor ( B ) denote the cofactor matrix of B . We pull out a factor α i and evaluate det X S along its i th column. This gives us

(6) det X { i } = α i ( 1 t C i ( Cofactor ( B ) ) ) .

It is easy to check that

(7) j th entry of C i ( Cofactor ( B ) ) = ( − 1 ) n − 2 ( − 2 ) n − 3 ∏ k ≠ j α k , when i ≠ j , ( − 2 ) n − 3 ( n − 4 ) ∏ k ≠ i α k , when i = j .

By (6) and (7), we have

(8) det X { i } = ( − 2 ) n − 3 ( n − 4 ) ∏ k = 1 n − 1 α k − α i 2 ∑ k = 1 k ≠ i n − 1 ∏ j = 1 j ∉ { i , k } n − 1 α j .

Substituting (8) and (5) into (4) and using (3), we obtain

det W K = det X ∅ + ∑ i = 1 n − 1 det X { i } = ( − 1 ) n − 2 2 n − 3 Γ ( T ) .

This completes the proof.□

Remark 9

By setting α i = 1 for 1 ≤ i < n , we obtain an unweighted tree T . In this case, we obtain WK = WtMin4PC T [ E , E ] = 2 ( J − I ) , and det W K = ( − 1 ) n − 2 2 n − 1 ( n − 2 ) . Thus, Lemma 8 generalizes [13, Lemma 3].

Lemma 10

Let T be an edge-weighted tree on n vertices with edge set E = { e 1 , … , e n − 1 } . Let WK = WtMin4PC T [ E , E ] . Define a vector z ∈ R n − 1 , with its coordinates indexed by E. Denote by z ( e i ) , the coordinate of z indexed by e i , and define it as

z ( e i ) = ∏ j = 1 , j ≠ i n − 1 α j ∑ j = 1 , j ≠ i n − 1 α j − ( n − 4 ) α i .

Then, WK z = Γ ( T ) w , where w t = [ α 1 , … , α n − 1 ] t .

Proof

The result follows from the fact that

□ ∑ j = 1 n − 1 ∏ k ≠ j α k ∑ k ≠ j α k = ∑ j = 1 n − 1 ∑ S ⊂ [ n − 1 ] \ { j } ∣ S ∣ = n − 3 ( α j 2 ∏ k ∈ S α k ) .

For a tree T on n vertices, Γ ( T ) as described in (3) can also be expressed as the following.

(9) Γ ( T ) = ∑ i = 1 n − 1 ( Sum T ( α ) − α i ) Prod T ( α ) α i − ( n − 2 ) ( n − 5 ) Prod T ( α ) .

Lemma 11

Let T be a tree on n ≥ 2 vertices with edge set E = { e 1 , … , e n − 1 } . Then, Γ ( T ) ≠ 0 , and thus, WtMin4PC T [ E , E ] is invertible.

Proof

By (9), it is easy to see that

Γ ( T ) = Sum T ( α ) Prod T ( α ) ∑ i = 1 n − 1 1 α i − ( n − 3 ) 2 Prod T ( α ) .

As α i for 1 ≤ i ≤ n − 1 are positive, by the AM ≥ HM inequality, we obtain

Sum T ( α ) ∑ i = 1 n − 1 1 α i ≥ ( n − 1 ) 2 .

Hence, Γ ( T ) > 0 when n > 2 . By Lemma 8, we have det WtMin4PC T [ E , E ] ≠ 0 . This completes the proof.□

Let T be a tree on n vertices. Let i and j be two nonadjacent vertices in T and e = { i , j } . In our next result, we analyse the result of Lemma 10 carefully and find the unique solution x to the linear system of equations WK x = Γ ( T ) WtMin4PC T [ E , e ] .

Lemma 12

Let T be a tree on n vertices with edge set E = { e 1 , … , e n − 1 } . For 1 ≤ i ≤ n − 1 , let e i have weight α i . Let WK = WtMin4PC T [ E , E ] and e = { i , j } ∉ E . Suppose that z ∈ R n − 1 is the vector defined in Lemma 10. We define a vector x e ∈ R n − 1 , with its coordinates indexed by E. For f ∈ E , denote the entry indexed by f in x e as x e ( f ) , and define it by

x e ( f ) = − ( d i , j − 1 ) z ( f ) + Γ ( T ) , if f i s o n t h e i j - p a t h , − ( d i , j − 1 ) z ( f ) , if f i s n o t o n t h e i j - p a t h .

Then, we have WK x e = Γ ( T ) WtMin4PC T [ E , e ] .

Proof

Note for i = 1 , … , n − 1 that

(10) WtMin4PC T ( e i , e ) = w i , j − α i , if e i is on the i j -path , w i , j + α i , if e i is not on the i j -path .

Recall e = { i , j } ∉ E . Define γ e ∈ R n − 1 as the vector with its r th coordinate being 1 if the edge e r is on the i j -path and 0 otherwise. For e r ∈ E , it is easy to note that

(11) ( WK γ e ) ( e r ) = w i , j + ( d i , j − 2 ) α r , if e r is on the i j -path , w i , j + d i , j α r , if e r is not on the i j -path .

By Lemma 10, for e r ∈ E , we have

(12) ( WK x e ) ( e r ) = − ( d i , j − 1 ) Γ ( T ) α r + Γ ( T ) ( WK γ e ) ( e r ) .

Therefore, the result follows by substituting (11) into (12) and using the identity (10).□

We will need the following result to determine both the rank of WtMin4PC T and the determinant of WtMin4PC T [ B , B ] , where B is a basis of the row space of WtMin4PC T .

Lemma 13

Let T be a tree with n vertices with edge set E. For 1 ≤ i ≤ n − 1 , let edge e i have weight α i . Let a = { i , j } ∉ E and b = { k , ℓ } ∉ E . Let z and x a be the vectors defined Lemmas 10 and 12, respectively. Then,

(13) w t z = 2 Sum T ( α ) Prod T ( α ) ,

(14) 1 t z = Γ ( T ) − 2 ( n − 3 ) Prod T ( α ) , and

(15) WtMin4PC T [ a , E ] x b = Γ ( T ) WtMin4PC T ( a , b ) + 2 ( d i , j − 1 ) ( d k , ℓ − 1 ) Sum T ( α ) Prod T ( α ) ,

where WtMin4PC T [ a , E ] is a submatrix of WtMin4PC T restricted to the row indexed by a and the columns indexed by E.

Proof

It is easy to see that (13) follows from

∑ e i ∈ E ∑ e j ∈ E \ { e } α j − ( n − 4 ) α i = 2 Sum T ( α ) .

Furthermore, by (9), we have

1 t z = ∑ j = 1 n − 1 ∏ k ≠ j α k ∑ r ≠ j α r − ( n − 4 ) ( n − 1 ) Prod T ( α ) = Γ ( T ) − 2 ( n − 3 ) Prod T ( α ) .

Let P a and P b denote the i , j -path and k , ℓ -path, respectively, in T . Define the vector τ a ∈ R n − 1 with entries indexed by E as follows: τ a ( e r ) = − α r if e r ∈ P a and τ a ( e r ) = α r if e r ∉ P a . Define τ b in an identical manner using P b instead of P a . Note that

τ a t z = − ∑ e r ∈ P a α r ∏ j = 1 , j ≠ r n − 1 α j ∑ k = 1 , k ≠ r α k + ∑ e s ∉ P a α s ∏ j = 1 , j ≠ s n − 1 α j ∑ k = 1 , k ≠ s n − 1 α k + ( n − 4 ) Prod T ( α ) ∑ e s ∉ P a α s − ∑ e r ∈ P a α r = Prod T ( α ) ∑ e s ∉ P a ∑ j = 1 , j ≠ s n − 1 α j − ∑ e r ∈ P a ∑ k = 1 , k ≠ r n − 1 α k + ( n − 4 ) Prod T ( α ) ∑ e s ∉ P a α s − ∑ e r ∈ P a α r = Sum T ( α ) Prod T ( α ) ( n − 1 − 2 d i , j ) − ( n − 3 ) Prod T ( α ) ∑ e s ∉ P a α s − ∑ e r ∈ P a α r = Sum T ( α ) Prod T ( α ) ( n − 1 − 2 d i , j ) − ( n − 3 ) Sum T ( α ) Prod T ( α ) + 2 ( n − 3 ) Prod T ( α ) w i , j = − 2 ( d i , j − 1 ) Sum T ( α ) Prod T ( α ) + 2 ( n − 3 ) Prod T ( α ) w i , j .

By (10), we have WtMin4PC T [ E , a ] = w i , j 1 + τ a . Note that x b = − ( d k , ℓ − 1 ) z + Γ ( T ) γ b , where γ b is the vector defined in the proof of Lemma 12. Clearly, 1 t γ b = d k , ℓ . Thus, it follows that

WtMin4PC T [ a , E ] x b = ( w i , j 1 t + τ a t ) ( − ( d k , ℓ − 1 ) z + Γ ( T ) γ b ) = − w i , j ( d k , ℓ − 1 ) Γ ( T ) + 2 ( n − 3 ) w i , j ( d k , ℓ − 1 ) Prod T ( α ) + Γ ( T ) d k , ℓ d i , j + Γ ( T ) τ a t γ b + 2 ( w i , j − 1 ) ( d k , ℓ − 1 ) Sum T ( α ) Prod T ( α ) − 2 ( n − 3 ) ( d k , ℓ − 1 ) w i , j Prod T ( α ) = Γ ( T ) ( w i , j + τ a t γ b ) + 2 ( d i , j − 1 ) ( d k , ℓ − 1 ) Sum T ( α ) Prod T ( α ) .

Let E ( P a ) and E ( P b ) be the set of edges in the paths P a and P b , respectively. Note that

τ a t γ b = ∑ e r ∈ E ( P b ) \ E ( P a ) α r − ∑ e s ∈ E ( P a ) ∩ E ( P b ) α s .

It follows that

(16) w i , j + τ a t γ b = ∑ e r ∈ E ( P a ) \ E ( P b ) α r + ∑ e s ∈ E ( P b ) \ E ( P a ) α s .

Furthermore, note that

(17) WtMin4PC T ( a , b ) = w i , j + w k , ℓ , if E ( P a ) ∩ E ( P b ) = ∅ , w i , j + w k , ℓ − 2 ∑ e r ∈ E ( P a ) ∩ E ( P b ) α r , if E ( P a ) ∩ E ( P b ) ≠ ∅ .

By (16) and (17), we have WtMin4PC T ( a , b ) = τ a t γ b , completing the proof.□

Our next result generalizes [13, Theorem 9], and it shows that the rank of WtMin4PC T is n , showing that it is independent of the structure of T and its edge weights.

Theorem 14

Let T be a tree on n > 2 vertices with edge set E. Then, the rank of WtMin4PC T is n .

Proof

Let E c be the set of edges in T c and define the three matrices WK = WtMin4PC T [ E , E ] , WK 1 = WtMin4PC T [ E , E c ] and WK 2 = WtMin4PC T [ E c , E c ] . Note that WtMin4PC T can be written in block form as

WtMin4PC T = WK WK 1 WK 1 t WK 2 .

Lemma 11 shows that det WK ≠ 0 . Applying the Guttman rank additivity formula^[1] [19, Equation (0.9.2)], we obtain

(18) rank ( WtMin4PC T ) = rank ( WK ) + rank ( WK 2 − WK 1 t WK − 1 WK 1 ) .

Let a = { i , j } ∉ E and b = { k , ℓ } ∉ E . Let WtMin4PC T [ E , a ] be the submatrix of WtMin4PC T restricted to the rows indexed by E and the column indexed by a . Hence, WtMin4PC T [ E , a ] is an ( n − 1 ) -dimensional column vector. In an identical manner, define WtMin4PC T [ E , b ] . Denote by Col a ( WK 1 ) = WtMin4PC T [ E , a ] and Col b ( W K 1 ) = WtMin4PC T [ E , b ] the two columns of WK 1 indexed by a and b respectively. Let x a and x b be vectors defined in Lemma 12 that are indexed by a and b respectively. Using Lemma 12, it is simple to check that WK − 1 Col a ( W K 1 ) = 1 Γ ( T ) x a . Therefore, by Lemma 13,

(19) Col b ( WK 1 ) t WK − 1 Col a ( WK 1 ) = WtMin4PC T ( b , a ) + ( d i , j − 1 ) ( d k , ℓ − 1 ) X ,

where X = Sum T ( α ) Prod T ( α ) ∕ Γ ( T ) .

Define a vector μ ∈ R n 2 − ( n − 1 ) whose coordinates are indexed by the element of E c as follows: for e = { s , t } ∉ E ( T ) define μ ( e ) = ( d s , t − 1 ) X . It is easy to note that

WK 2 − WK 1 t WK − 1 WK 1 = μμ t .

Using (18), our proof is complete.□

Corollary 15

We need (19) when a = b . We note that as follows:

Col a ( WK 1 ) t WK − 1 Col a ( WK 1 ) = ( d i , j − 1 ) 2 X .

In our last main result of this section, we describe a class B of bases for the row space of WtMin4PC T and also give a formula for the determinant of the matrix obtained by restricting WtMin4PC T to the elements with both row and column indices from B ∈ B .

Proof of Theorem 3

The result follows by first applying the Schur complement formula to the matrix WtMin4PC T [ B , B ] as done in (2), and then using Corollary 15 along with Lemmas 8 and 12.□

3 Rank of WtMax4PC T

In [1, Theorem 1], Azimi et al. showed for a tree T with n vertices and p leaves that rank ( Max4PC T ) = 2 ( n − p ) . In the course of their proof, they designed an algorithm that outputs a class B of bases for the row space of Max4PC T . In this section, we extend this result to weighted trees. This shows that the rank of WtMax4PC T is independent of the edge weights. Thus, our following result generalizes [1, Lemma 3] to weighted trees.

Lemma 16

Let T be a tree on n vertices. Let n − 1 , n ∈ V ( T ) be such that vertex n is a leaf in T with unique neighbour n − 1 . If u is a vertex of T other than n and n − 1 , then for each pair { i , j } of vertices, we have

WtMax4PC T ( { u , n } , { i , j } ) = WtMax4PC T ( { u , n − 1 } , { i , j } ) + w n − 1 , n .

Proof

Denote w n − 1 , n = α for brevity, and recall u ≠ n − 1 , n . Clearly, if v ∈ V ( T ) with v ≠ n , then the v - n path must include the vertex n − 1 . We thus have

(20) w v , n = w v , n − 1 + α and w u , n = w u , n − 1 + α .

For each pair { i , j } of vertices in T , by the definition of the matrix WtMax4PC T , we obtain

(21) WtMax4PC T ( { u , n } , { i , j } ) = max { w u , n − 1 + w i , j + α , w u , i + w n , j , w u , j + w n , i } .

We split our proof into two cases: when both i and j are different from n and when either i = n or j = n .

Case 1: This is the case of i ≠ n and j ≠ n . Substituting (20) into (21), we obtain

WtMax4PC T ( { u , n } , { i , j } ) = max { w u , n − 1 + w i , j + α , w u , i + w n − 1 , j + α , w u , j + w n − 1 , i + α } = WtMax4PC T ( { u , n − 1 } , { i , j } ) + α .

Our proof is complete in this case.

Case 2: This is the case of i = n or j = n . Without loss of generality, assume j = n , and thus, i ≤ n − 1 . By the triangle inequality, we have

(22) w u , n − 1 + w i , n − 1 ≥ w u , i and w u , n + w i , n ≥ w u , i + 2 α .

By (22), the entry of WtMax4PC T indexed by the row { u , n } and the column { i , n } is

WtMax4PC T ( { u , n } , { i , n } ) = max { w u , n + w i , n , w u , i } = w u , n + w i , n .

Furthermore, if k = WtMax4PC T ( { u , n − 1 } , { i , n } ) , then we have

k = max { w u , n − 1 + w i , n , w u , i + α , w u , n + w n − 1 , i } = max { w u , n + w i , n − α , w u , i + α , w u , n + w n , i − α } [ by (20) ] = w u , n + w i , n − α [ by (20) and (22) ] = WtMax4PC T ( { u , n } , { i , n } ) − α .

Thus, we have proved the result for the second case. Our proof is complete.□

Our next lemma extends [1, Lemma 4] to weighted trees.

Lemma 17

Let T be a tree on n vertices. Let y , z ∈ V ( T ) be such that y is a leaf and z is the quasi-pendant adjacent to y. Let u ∈ V ( T ) be a neighbour of z other than y and B u be the connected component of T − z that contains the vertex u (Figure 1). Then,

WtMax4PC T ( { y , z } , { i , j } ) = WtMax4PC T ( { u , z } , { i , j } ) + w y , z + w u , z , if i , j ∈ B u , WtMax4PC T ( { u , z } , { i , j } ) + w y , z − w u , z , otherwise .

Figure 1

Illustrating the setup for Lemma 17.

Proof

Let i and j be two distinct vertices of T with j ∈ B u . Using triangle inequality, it is simple to see that

(23) w i , j ≤ w u , i + w z , j − w u , z .

We first assume i , j ∈ B u ∩ V ( T ) . Clearly, w y , v = w u , v + w y , z + w u , z for each v ∈ B u . Therefore, we have

WtMax4PC T ( { y , z } , { i , j } ) = max { w y , z + w i , j , w y , i + w z , j , w y , j + w z , i } = max { w y , z + w i , j , w u , i + w z , j + w y , z + w u , z , w u , j + w z , i + w y , z + w u , z } = max { w i , j − w u , z , w u , i + w z , j , w u , j + w z , i } + w y , z + w u , z = max { w u , z + w i , j , w u , i + w z , j , w u , j + w z , i } + w y , z + w u , z [ by (23) ] = WtMax4PC T ( { u , z } , { i , j } ) + w y , z + w u , z .

Now, consider the case when i ∉ B u and j ∈ V ( T ) . First, note that if i = y and j ∈ V ( T ) − { y } , then w y , j + w u , z = w z , j + w y , u . Combining these with w z , j ≤ w y , j gives

WtMax4PC T ( { y , z } , { y , j } ) = max { w y , z + w y , j , w z , j , w y , j + w y , z } = max { w u , z + w y , j + w y , z − w u , z , w y , j + w z , u + w y , z − w u , z } = max { w u , z + w y , j , w y , u + w z , j , w y , j + w z , u } + w y , z − w u , z = WtMax4PC T ( { u , z } , { y , j } ) + w y , z − w u , z .

We split the remaining part of our proof into two cases when i ∉ B u ∪ { y } with j ∈ B u and when i ∉ B u ∪ { y } with j ∉ B u .

Case 1: This is the case of i ∉ B u ∪ { y } and j ∈ B u . In this case, it is easy to see that w i , j = w i , z + w z , u + w u , j , and so, we obtain

(24) w u , i + w z , j = w i , j + w u , z = w u , j + w z , i + 2 w u , z ≥ w i , j − w u , z .

Therefore,

WtMax4PC T ( { y , z } , { i , j } ) = max { w y , z + w i , j , w y , i + w z , j , w y , j + w z , i } [ by w y , i + w u , z = w u , i + w y , z ] = max { w y , z + w i , j , w u , i + w z , j + w y , z − w u , z , w u , j + w z , i + w y , z + w u , z } = max { w u , z + w i , j , w u , i + w z , j , w u , j + w z , i } + w y , z − w u , z [ by (24) ] = WtMax4PC T ( { u , z } , { i , j } ) + w y , z − w u , z .

Case 2: This is the case of i ∉ B u ∪ { y } and j ∉ B u . In this case, note that if j ≠ y , then w y , i + w u , z = w u , i + w y , z and w y , j + w u , z = w u , j + w y , z . Using these, it is easy to see that

WtMax4PC T ( { y , z } , { i , j } ) = max { w y , z + w i , j , w y , i + w z , j , w y , j + w z , i } = max { w u , z + w i , j + w y , z − w u , z , w u , i + w z , j + w y , z − w u , z , w u , j + w z , i + w y , z − w u , z } = WtMax4PC T ( { u , z } , { i , j } ) + w y , z − w u , z .

Finally, if we have j = y , then i ∉ B u ∪ { y } . In this case, one can check that w y , i = w u , i + w y , z − w u , z . Using this, we obtain

WtMax4PC T ( { y , z } , { i , y } ) = max { w y , z + w i , y , w y , i + w y , z , w z , i } = max { w u , z + w i , y + w y , z − w u , z , w y , i + w y , z + w y , z − w u , z } [ as w z , i + w y , z = w y , i ] = max { w u , z + w i , y , w z , i + w u , y , w u , i + w y , z } + w y , z − w u , z = WtMax4PC T ( { u , z } , { i , y } ) + w y , z − w u , z .

This completes the proof.□

The next result is our main result for this section.

Theorem 18

Let T be a tree on n ≥ 3 vertices with p leaves. Then, rank ( WtMax4PC T ) = 2 ( n − p ) .

Proof

We induct on n , the number of vertices in T . Our base case is when n = 3 . The only possible tree is P 3 , the path tree on three vertices. Let the two edges of P 3 have weights α 1 and α 2 , respectively. Then, it can be verified that

WtMax4PC P 3 = 2 α 1 2 α 1 + α 2 α 1 + α 2 2 α 1 + α 2 2 α 1 + 2 α 2 α 1 + 2 α 2 α 1 + α 2 α 1 + 2 α 2 2 α 2 .

It is easy to verify that rank ( WtMax4PC P 3 ) = 2 . Thus, our base case is done.

Thus, assume that the result is true for all trees with n − 1 vertices. Let T be a tree on n vertices. Without loss of any generality, we assume that n is a leaf adjacent to n − 1 . Let T ^ = T − { n } be the tree obtained from T by deleting the vertex n from T . We divide the proof into two cases based on the degree of vertex n − 1 in T .

Case 1: This is the case when there exists a quasi pendant vertex n − 1 with degree two. Let n − 2 be the neighbour of n − 1 other than n . Let w n − 1 , n = α n − 1 and w n − 2 , n − 1 = α n − 2 .

Let V n be the collection of 2-sized subsets of [ n ] = { 1 , 2 , … , n } and U n − 1 = { { i , n } ∣ i ∈ [ n − 1 ] } . For listing the rows and columns of WtMax4PC T , we order the elements of V n as V n = ( V n − 1 , U n − 1 ) . For R , C ⊆ V n , let WtMax4PC [ R , C ] denote the submatrix of WtMax4PC T induced on the rows in R and the columns in C . With this notation, let WtMax4PC 12 = WtMax4PC T [ V n − 1 , U n − 1 ] and WtMax4PC 22 = WtMax4PC T [ U n − 1 , U n − 1 ] . Then, WtMax4PC T can clearly be written in the following partitioned form:

WtMax4PC T = WtMax4PC T ^ WtMax4PC 12 WtMax4PC 12 t WtMax4PC 22 .

In the matrix WtMax4PC T , perform the elementary row operation Row u , n = Row u , n − Row u , n − 1 and the column operation Col u , n = Col u , n − Col u , n − 1 for 1 ≤ u ≤ n − 1 . Denoting equivalent matrices by A ∼ B , Lemma 16 gives us

WtMax4PC T ∼ WtMax4PC T ^ α n − 1 J n − 1 2 × ( n − 2 ) u α n − 1 J ( n − 2 ) × n − 1 2 0 ( n − 2 ) × ( n − 2 ) α n − 1 1 u t α n − 1 1 t 2 α n − 1 . WtMax4PC T ∼ WtMax4PC T ^ 0 ⋯ 0 α n − 1 ⋮ ⋱ ⋮ ⋮ 0 ⋯ 0 α n − 1 u 0 ⋯ 0 ⋮ ⋱ ⋮ 0 ⋯ 0 α n − 1 ⋯ α n − 1 0 0 ⋮ 0 α n − 1 u t 0 ⋯ 0 α n − 1 2 α n − 1 ,

where in the second step above, we perform the row operations Row u , n = Row u , n − Row n − 2 , n − 1 and identical column operations Col u , n = Col u , n − Col n − 2 , n − 1 for 1 ≤ u ≤ n − 2 . Recall that the vertex n − 1 is a leaf in the tree T ^ . Denote the row of a matrix M indexed by r as M [ r , : ] . Thus, by Lemma 16, for all v ∈ T ^ − { n − 1 , n − 2 } , we obtain

(25) WtMax4PC T ^ ( { n − 1 , v } , : ) = WtMax4PC T ^ ( { n − 2 , v } , : ) + α n − 2 1 t .

Let n − 3 be a neighbour of n − 2 in T and let w n − 3 , n − 2 = α n − 3 . By the definition of WtMax4PC T , it is simple to see that

(26) WtMax4PC T ( { n − 1 , n } , { n − 1 , n − 3 } ) = α n − 1 + α n − 2 + α n − 3 and

(27) WtMax4PC T ( { n − 1 , n } , { n − 2 , n − 3 } ) = α n − 1 + 2 α n − 2 + α n − 3 .

We perform the row operation Row n − 2 , n = α n − 2 Row n − 2 , n − α n − 1 ( Row n − 1 , n − 3 − Row n − 2 , n − 3 ) and an identical column operation Col n − 2 , n = α n − 2 Col n − 2 , n − α n − 1 ( Col n − 1 , n − 3 − Col n − 2 , n − 3 ) . Using (25)–(27) and the above row and column operation, we obtain the following matrix that is row and column equivalent to WtMax4PC T .

WtMax4PC T ∼ WtMax4PC T ^ 0 ⋯ 0 0 ⋮ ⋱ ⋮ ⋮ 0 ⋯ 0 0 u 0 ⋯ 0 ⋮ ⋱ ⋮ 0 ⋯ 0 0 ⋯ 0 0 0 ⋮ 0 2 α n − 1 α n − 2 u t 0 ⋯ 0 2 α n − 1 α n − 2 2 α n − 1 ∼ WtMax4PC T ^ 0 0 0 0 0 0 0 2 α n − 1 α n − 2 0 0 2 α n − 1 α n − 2 2 α n − 2 ∼ WtMax4PC T ^ 0 0 0 0 0 0 0 0 0 − 2 α n − 1 2 α n − 2 0 0 0 0 2 α n − 2 .

This completes the proof of Case 1.

Case 2: This is the case when any quasi-pendant vertex in T has degree at least three. Let [ v 1 , v 2 , … , v k ] be a maximum length path in T . Hence, its length equals the diameter of T . Clearly, v 1 is a pendant vertex and v 2 is a quasi-pendant vertex in T . Since the degree of v 2 in T is at least three, there is another leaf y with y being adjacent to v 2 and y ≠ v 1 . By relabelling, we assume that n and n − 2 are two leaves in T that are both adjacent to n − 1 . Let w n − 1 , n = α n − 1 and w n − 2 , n − 1 = α n − 2 . When 1 ≤ i < j ≤ n and u ∉ { n , n − 1 } , Lemma 16 gives

(28) WtMax4PC T ( { u , n } , { i , j } ) = WtMax4PC T ( { u , n − 2 } , { i , j } ) + α n − 1 and

(29) WtMax4PC T ( { u , n − 1 } , { i , j } ) = WtMax4PC T ( { u , n − 2 } , { i , j } ) + α n − 2 .

Let vertex n − 3 be adjacent to n − 2 in T , and let w n − 3 , n − 2 = α n − 3 . Let B n − 3 be the connected component of T − { n − 2 } that contains the vertex n − 3 . By Lemma 17, we have

(30) WtMax4PC T ( { n − 2 , n } , { i , j } ) = WtMax4PC T ( { n − 3 , n − 2 } , { i , j } ) + α n − 1 + α n − 3 , if i , j ∈ B n − 3 WtMax4PC T ( { n − 3 , n − 2 } , { i , j } ) + α n − 1 − α n − 3 , otherwise .

(31) WtMax4PC T ( { n − 2 , n − 1 } , { i , j } ) = WtMax4PC T ( { n − 3 , n − 2 } , { i , j } ) + α n − 2 + α n − 3 , if i , j ∈ B n − 3 WtMax4PC T ( { n − 3 , n − 2 } , { i , j } ) + α n − 2 − α n − 3 , otherwise .

Using (28) and (29), we obtain Row n − 3 , n − 1 − Row n − 3 , n − 2 = α n − 2 1 t and combining this with (30) and (31), we obtain ( Row n − 2 , n − Row n − 3 , n − 2 ) − ( Row n − 2 , n − 1 − Row n − 3 , n − 2 ) = ( α n − 1 − α n − 2 ) 1 t . Furthermore, for all i ∈ V ( T ) with i ∉ { n , n − 2 } , by (28)–(31), it is simple to check the following:

α n − 2 Row i , n = α n − 2 ( Row i , n − Row i , n − 2 ) − α n − 1 ( Row n − 3 , n − 1 + Row n − 3 , n − 2 ) and α n − 2 Col i , n = α n − 2 ( Col i , n − Col i , n − 2 ) − α n − 1 ( Col n − 3 , n − 1 + Col n − 3 , n − 2 ) .

Furthermore, note that

α n − 2 Row n − 2 , n = α n − 2 ( Row n − 2 , n − Row n − 3 , n − 2 ) − α n − 2 ( Row n − 2 , n − 1 − Row n − 3 , n − 2 ) − ( α n − 1 − α n − 2 ) ( Row n − 3 , n − 1 − Row n − 3 , n − 2 ) and α n − 2 Col n − 2 , n = α n − 2 ( Col n − 2 , n − Col n − 3 , n − 2 ) − α n − 2 ( Col n − 2 , n − 1 − Col n − 3 , n − 2 ) − ( α n − 1 − α n − 2 ) ( Col n − 3 , n − 1 − Col n − 3 , n − 2 ) .

Using the aforementioned row operations and the identical column operations, it is easy to see that the matrix WtMax4PC T is similar to the following block matrix:

WtMax4PC T ^ 0 0 0 .

Thus, Case 2 is proved. This completes our proof.□

The following result is an easy consequence of Theorem 18.

Corollary 19

Let T be a tree on n vertices with n > 3 . Suppose there exist two leaves u and v adjacent to the same vertex. Then, we have

rank ( WtMax4PC T ) = rank ( WtMax4PC T − u ) = rank ( WtMax4PC T − v ) .

4 Basis for the row space of WtMax4PC T and determinant

Let T be a weighted tree on n vertices with α i being the weight on edge e i for 1 ≤ i ≤ n − 1 . Let WtMax4PC T be its weighted maximum-4PC matrix. When α i = 1 for 1 ≤ i ≤ n − 1 , in [1, Lemma 9], Ali et al. constructed a class B of bases for the row space of Max4PC T . They also gave a formula for the determinant of Max4PC T [ B , B ] for all B ∈ B . As we will compare our weighted version with the unweighted version, we quote their unweighted result below.

Theorem 20

[1, Lemma 9 and Theorem 11] Let T be a tree on n vertices, and let Max4PC T be its maximum-4PC matrix. Let B be the class of bases given in [1, Lemma 9] for the row space of Max4PC T . If p is the number of leaves in T, then for all B ∈ B , ∣ B ∣ = 2 ( n − p ) and det Max4PC T [ B , B ] = ( − 1 ) n − p 2 2 ( n − p − 1 ) .

For 1 ≤ i ≤ n − 1 , if α i > 0 , we first show that each B ∈ B from Theorem 20, is a basis for the row space of WtMax4PC T . For proving Theorem 20, an algorithm inspired by depth first search was given. The output of this algorithm was shown to be a basis for all trees T other than the star trees. Star trees were handled separately. As there are some choices in listing adjacent blocks, the output of our algorithm is not unique, and so we obtain a class B with all B ∈ B being a basis. Thus, B is a class of bases that our algorithm outputs.

We move to our edge-weighted version and again separate star trees from the rest. Let S n denote the star tree on n vertices. We start by giving an identical class of bases for the row space of WtMax4PC S n . We denote the degree of the vertex u in T as deg ( u ) .

Theorem 21

Let S n be the weighted star tree on vertex set [ n ] and let deg ( 1 ) = n − 1 . For 1 < i ≤ n , let α i be the weight on the edge { 1 , i } . Then, rank ( WtMax4PC S n ) = 2 and for i , j with 1 < i < j ≤ n , the set of rows indexed by { 1 , i } and { i , j } forms a basis B for the row space of WtMax4PC S n . Denote by B , the class of such B over the choices of i , j . Furthermore, det WtMax4PC S n [ B , B ] = − α j 2 for all B ∈ B .

Proof

From Corollary 19, it follows when n ≥ 3 that rank ( WtMax4PC S n ) = rank ( WtMax4PC S 3 ) = 2 . To see this, note that

WtMax4PC S 3 = { 1 , 2 } { 1 , 3 } { 2 , 3 } { 1 , 2 } { 1 , 3 } { 2 , 3 } 2 α 2 α 2 + α 3 2 α 2 + α 3 α 2 + α 3 2 α 3 α 2 + 2 α 3 2 α 2 + α 3 α 2 + 2 α 3 2 α 2 + 2 α 3 ,

and α 2 × ( Row { 2 , 3 } − Row { 1 , 2 } ) = α 3 × ( Row { 2 , 3 } − Row { 1 , 3 } ) , where Row { i , j } is the row of WtMax4PC S 3 indexed by the pair { i , j } . When the tree is S n , one can verify that α i × ( Row { i , j } − Row { 1 , i } ) = α j ( Row { i , j } − Row { 1 , j } ) for i , j with 1 < i < j ≤ n . Furthermore, if B = { { 1 , i } , { i , j } } for some i , j ∈ S n , then det WtMax4PC S n [ B , B ] = − α j 2 ≠ 0 whenever α j ≠ 0 . Thus, the rows indexed by { 1 , i } and { i , j } form a basis for the row space of WtMax4PC S n . This completes the proof.□

One can ask when the rows indexed by { 1 , i } and { j , k } would be linearly independent.

Remark 22

For 1 < i ≤ j < k ≤ n , the rows indexed by { 1 , i } and { j , k } are linearly independent if α i ≠ α j + α k , since each 2 × 2 minor with rows and columns indexed by { 1 , i } and { j , k } has a factor ( α j + α k − α i ) .

Setting α i = 1 for 1 ≤ i ≤ n − 1 , it is easy to see that [1, Lemma 7] is a direct consequence of Theorem 21.

Corollary 23

[1, Lemma 7] Let S n be the star tree on n vertices with deg ( 1 ) = n − 1 . Then, rank ( Max4PC S n ) = 2 and for 1 < i < j ≤ n , the rows indexed by { 1 , i } and { i , j } are linearly independent. Let B the class of bases defined in Theorem 21. Then, for each B ∈ B , we have det Max4PC S n [ B , B ] = − 1 .

We introduce a term “pendant edge” for our next result. In a tree T , an edge is known as a pendant edge if one of its end points is a leaf (i.e., one of its end points is a pendant vertex). If both end points of an edge e have degree strictly greater than one, then such an e will be called as a non-pendant edge. In our next result, we show that the set B output by an algorithm described in [1, Theorem 20] is a basis for the row space of WtMax4PC T as well.

Theorem 24

Let T be a tree on n > 3 vertices with p leaves. We assume that T ≠ S n and for 1 ≤ i ≤ n − 1 , edge e i has weight α i . Let B be the collection of bases as described in [1, Theorem 20]. For each B ∈ B , the following statements are true.

There exist a unique pendant edge e = { u , v } ∈ B ∩ E ( T ) such that deg ( u ) = 1 with { u , w } ∈ B , where w is a neighbour of v and deg ( w ) > 1 . Moreover, if ℓ ≠ u is a leaf in T and { x , ℓ } ∈ B ∩ E ( T ) for some x ∈ T , then { x , y } ∈ B ∩ E ( T ) for some neighbour y of x with deg ( y ) ≠ 1 .
The number of elements in B is 2 ( n − p ) .
The set B forms a basis for the row space of WtMax4PC T .

Figure 2

Illustrative figure for the argument above.

Proof

The proofs of items (a) and (b) do not involve weights on the edges and so directly follow from [1, Theorem 11].

Proof of Item (c). We induct on n . When n = 4 , if we exclude the star tree, we are left with P 4 , the path on 4 vertices. Thus, our base case is P 4 . Suppose P 4 has edges { { 1 , 2 } , { 2 , 3 } , { 3 , 4 } } with respective weights α 1 , α 2 and α 3 . If we have B = { { 1 , 2 } , { 1 , 3 } , { 2 , 3 } , { 3 , 4 } } ∈ B , then it can easily be verified that det WtMax4PC P 4 [ B , B ] = 4 α 1 2 α 2 2 ≠ 0 . Thus, from Theorem 18, the result is true for our base case.

We now assume that the result holds for all trees on n − 1 vertices. Let tree T ≠ S n be a tree on n vertices with edge set E ( T ) . Let B be a collection of sets output by the algorithm described in Theorem 20, and let B ∈ B . From part (a), there exist unique triplet of vertices u , v , and w in T such that { u , v } , { u , w } ∈ B with deg ( u ) = 1 and { u , v } , { v , w } ∈ E ( T ) . Without loss of generality, we assume that u = 1 , v = 2 , and w = 3 . Let the weight on the edge { 1 , 2 } be α 1 . Therefore from Lemma 16, one can see that

(32) WtMax4PC T [ B , { 1 , 3 } ] = WtMax4PC T [ B , { 2 , 3 } ] + α 1 1 .

From Item (a), recall that if ℓ ≠ 1 is a leaf in T and { v 1 , ℓ } ∈ B for some v 1 ∈ T , then it follows that { v 1 , ℓ } ∈ E ( T ) and { v 1 , v 2 } ∈ B for some neighbour v 2 of v 1 other than ℓ . Without loss of generality, assume that x is a leaf located on a path whose length is the diameter of T with { x , y } ∈ E ( T ) for some y ∈ T . Note that if there are multiple leaves attached to y , then by the induction hypothesis and applying Corollary 19, it follows that B is indeed a basis for the row space of WtMax4PC T . We next assume that deg ( y ) = 2 and { y , z } ∈ E ( T ) with { x , y } , { y , z } ∈ B and with z ≠ x . Furthermore, let w be the neighbour of z (other than y ) such that { z , w } ∈ B (Figure 2).

Consider the set B ^ obtained by applying our algorithm as described in Theorem 20 on T − x that makes the same choices, which resulted in the set B . By the induction hypothesis, B ^ is a basis for the row space of WtMax4PC T − x . Suppose the weights on { x , y } , { y , z } , and { z , w } are α , β and γ , respectively. Now, we divide the remaining part of the proof into two cases depending on whether { y , z } ∈ B ^ or { y , z } ∉ B ^ .

Case 1: This is the case of { y , z } ∈ B ^ . From our algorithm given in Theorem 20, it is simple to check that { y , w } ∈ B . Thus, B ^ = B \ { { x , y } , { y , w } } . If u = WtMax4PC T [ B ^ , { x , y } ] and v = WtMax4PC T − x [ B ^ , { y , w } ] are the column vectors of WtMax4PC T [ B ^ , B ^ ] indexed by the columns { x , y } and { y , w } , respectively, then, it is easy to see that the matrix WtMax4PC T [ B , B ] has the following partitioned form:

WtMax4PC T [ B , B ] = { x , y } { y , w } { x , y } { y , w } WtMax4PC T − x [ B ^ , B ^ ] u u u t 2 α α + β + γ v t α + β + γ 2 ( β + γ ) .

From Lemma 16, one can see that

(33) WtMax4PC T − x [ B ^ , { y , w } ] = WtMax4PC T − x [ B ^ , { z , w } ] + β 1 .

Furthermore, it is easy to see that

WtMax4PC T ( { z , w } , { x , y } ) = α + 2 β + γ and WtMax4PC T ( { z , w } , { y , w } ) = 2 β + γ .

Perform the row operation Row { y , w } = Row { y , w } − Row { z , w } − β α 1 ( Row { 1 , 3 } − Row { 2 , 3 } ) and an identical column operation in WtMax4PC T [ B , B ] . Using (32) and (33) in addition, we obtain the following:

{ x , y } { y , w } { x , y } { y , w } WtMax4PC T − x [ B ^ , B ^ ] u 0 u t 2 α − 2 β 0 t − 2 β 0 .

It follows that det WtMax4PC T [ B , B ] = − 4 β 2 det WtMax4PC T − x [ B ^ , B ^ ] . Hence, using the induction hypothesis, B is a basis for the row space of WtMax4PC T .

Case 2: This is the case of { y , z } ∉ B ^ . Using our algorithm from Theorem 20, as done in Case 1, the basis B ^ for the row space of WtMax4PC T − x can be determined from B . We have B ^ = B \ { { y , z } , { x , y } } , and the matrix WtMax4PC T [ B , B ] can be partitioned as follows:

WtMax4PC T [ B , B ] = { y , z } { x , y } { y , z } { x , y } WtMax4PC T − x [ B ^ , B ^ ] w u w t 2 β α + β u t α + β 2 α ,

where u = WtMax4PC T [ B ^ , { x , y } ] and w = WtMax4PC T − x [ B ^ , { y , z } ] .

Since { y , z } ∉ B ^ , it follows that if i , j ∈ B z for each { i , j } ∈ B ^ . Thus, by applying Lemma 17, we have u = w + ( α + β ) 1 . Performing the row operation Row { x , y } = Row { x , y } − Row { y , z } − ( α + β ) α 1 ( Row { 1 , 3 } − Row { 2 , 3 } ) and an identical column operation in WtMax4PC T [ B , B ] give us the following equivalent form:

{ y , z } { x , y } { y , z } { x , y } WtMax4PC T − x [ B ^ , B ^ ] w 0 w t 2 β − 2 β 0 t − 2 β 0 .

Thus, det WtMax4PC T [ B , B ] = − 4 β 2 det WtMax4PC T − x [ B ^ , B ^ ] . By induction, we obtain that B is a basis for the row space of WtMax4PC T , completing the proof.□

We are finally ready to prove Theorem 4.

Proof of Theorem 4

We use induction on n , where again, our base case is when n = 4 and T = P 4 . Using the base case of our proof of Item (c) of Theorem 24, we have det WtMax4PC P 4 [ B , B ] = 4 α 1 2 α 2 2 , where α 1 and α 2 are weights of the unique pendant edge e ∈ B mentioned in Item (a) of Theorem 24 and of the non-pendant edge of P 4 respectively. Assume that the result is true for all trees with n − 1 vertices. Let T ≠ S n be a tree with n vertices. Then, the result follows from the proof of Item (c) of Theorem 24 and the induction hypothesis. The proof is complete.□

Note that Theorem 4 is also true for the star tree S n . Thus, combining Theorems 21 and 24, we obtain Theorem 20 by setting each weight α i = 1 for all i ≥ 1 .

ali.azimi61@gmail.com

Acknowledgements

Some results in this work were in their conjectural form programmed and tested using SageMath. We thank the authors of SageMath for generously releasing it as a free software. We also would like to thank the anonymous reviewers for their careful reading and valuable comments which made our article better.

Funding information: R. Jana expresses deep gratitude to the Indian Institute of Technology Bombay, India, for the financial support provided via the Institute Post-doctoral Fellowship.
Author contributions: The authors confirm sole responsibility for the following: manuscript preparation; study conception and design; data collection; analysis and interpretation of results.
Conflict of interest: The authors declare no conflicts of interest.
Data availability statement: No data were used for the research described in the article.

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Received: 2023-11-16

Revised: 2024-05-18

Accepted: 2024-05-19

Published Online: 2024-07-02

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https://doi.org/10.1515/spma-2024-0011

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distance matrix; tree; four-point condition; edge weights

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