Squared distance matrix of a tree R.B.Bapat Indian Statistical - - PowerPoint PPT Presentation

Squared distance matrix of a tree

R.B.Bapat Indian Statistical Institute New Delhi, India

Matrices associated with a graph

Adjacency matrix A Incidence matrix Q Laplacian matrix L = QQ′ Distance matrix D

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The distance matrix of the tree is given by D =           1 2 2 3 3 4 1 1 1 2 2 3 2 1 2 3 3 4 2 1 2 1 1 2 3 2 3 1 2 3 3 2 3 1 2 1 4 3 4 2 3 1          

Collaborators in the area of distance matrices: Lal, Pati, Kirkland, Neumann, Balaji, Chebotarev, Sivasubramanian, Gupta, Rekhi, Kurata

Theorem (Graham and Pollak, 1971)

Let T be a tree with n vertices and let D be the distance matrix of

• T. Then

detD = (−1)n−1(n − 1)2n−2.

Weighted version

Theorem

Let T be a tree with n vertices. Let the edges of the tree carry weights α1, . . . , αn−1. Let D be the (weighted) distance matrix of

• T. Then

detD = (−1)n−12n−2 n−1

• i=1

αi n−1

• i=1

αi.

Matrix weights

Theorem

Let T be a tree with n vertices. Let the edges of the tree carry weights A1, . . . , An−1, which are s × s matrices. Let D be the (weighted) distance matrix of T. Then the determinant of D equals (−1)(n−1)s2(n−2)sdet n−1

• i=1

Ai

• det

n−1

• i=1

Ai

• .

Example

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A

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B

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C

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Let A, B, C be s × s matrices. Then the determinant of     A A + B A + B + C A B B + C A + B B C A + B + C B + C C     is (−1)s22sdet(A + B + C)det(ABC)

Properties of the Laplacian L

L = QQ′ =      d1 · · · d2 · · · . . . . . . ... . . . · · · dn      − A L is symmetric, with zero row and column sums. L is positive semidefinite. If G is connected then rank of L is n − 1.

Notation and some identities

T : tree with n vertices {1, 2, . . . , n}, di : degree of i, τi = 2 − di δ =      d1 d2 . . . dn      , τ = 21 − δ =      τ1 τ2 . . . τn      Q′DQ = −2I, LDL = −2L, Dτ = (n − 1)1

Inverse of D

Theorem (Graham and Lov´ asz, 1978)

Let T be a tree with n vertices. Let D be the distance matrix and L the Laplacian of T. Then D−1 = −1 2L + 1 2(n − 1)ττ ′

Moore-Penrose inverse of the Laplacian

If the Laplacian L has the spectral decomposition L = λ1x1x′

1 + · · · + λn−1xn−1x′ n−1,

then L+ = 1 λ1 x1x′

1 + · · · +

1 λn−1 xn−1x′

n−1.

Distance in trees and Laplacian minors

Let T be a tree with Laplacian L. Then (i) d(i, j) = det(L(i, j|i, j)) (ii) Any principal minor of L = a constant × (sum of the cofactors in the complementary principal submatrix of D) (iii) d(i, j) = ℓ+

ii + ℓ+ jj − 2ℓ+ ij

(iv) 1′D1 = 2(Wiener index of T) = 2n(trace L+)

Inverse of the resistance matrix

Let G be a connected graph with Laplacian L. The resistance distance between vertices i, j is given by r(i, j) = det(L(i, j|i, j)) det(L(i|i))

Theorem

Let G be a connected graph with resistance matrix R and Laplacian matrix L. Then R−1 = −1 2L + a rank one matrix

A generalized Laplacian

We will refer to an n × n matrix as a Laplacian if it is symmetric, has row and column sums zero, and has rank n − 1. We seek formulae of the form (distance matrix)−1 = − 1

2 (Laplacian) + a rank one matrix

Distance matrix − → Laplacian

Let D be a symmetric, nonsingular matrix such that 1′D−11 = 0. Then L = 2D−111′D−1 1′D−11 − 2D−1 is a Laplacian. Hence we have D−1 = −1 2L + a rank one matrix

Laplacian − → Distance matrix

Let L be a Laplacian and define D = [d(i, j)] by d(i, j) = det(L(i, j|i, j)) det(L(i|i)) . Then D is nonsingular and D−1 = − 1

2L + a rank one matrix.

Inverse of a principal submatrix of the distance matrix

Theorem Let T be a tree with distance matrix D and Laplacian L. Let D = D11 D12 D21 D22

• , L =

L11 L12 L21 L22

• be a compatible partitioning. Then

D−1

11 = −1

2(L11 − L12L−1

22 L21) + a rank one matrix.

The result can be used to give a formula for the “terminal Wiener index” of a tree.

Schur complement in a rank one update

Lemma Let A be an n × n matrix. Then a Schur complement in (A + a rank one matrix) equals (a Schur complement in A) + a rank one matrix. The Lemma can be applied to D−1 = −1 2L + 1 2(n − 1)ττ ′ to get the previous result.

Euclidean distance matrix

Let p1, . . . , pn be points in Rk. The matrix with (i, j)-entry equal ||pi − pj||2 is called a Euclidean distance matrix. If the points are all on a sphere then the matrix is a spherical Euclidean distance matrix.

Inverse of a Euclidean distance matrix

Let D be a nonsingular, spherical Euclidean distance matrix with d(i, j) = ||pi − pj||2. Then 1′D−11 = 0 and D−1 = −1 2L + a rank one matrix, where L is a Laplacian such that p1, . . . , pn are the columns of (L+)

1 2 .

Remark

Distance matrix of a tree is a spherical Euclidean distance matrix. In particular, it has exactly one positive eigenvalue.

q-analogue

Let T be a tree with n vertices and let q be an indeterminate. If i, j are vertices of the tree T with d(i, j) = t, define the q-distance between i and j to be 1 + q + q2 + · · · + qt−1. (We set d(i, i) = 0.) Let Dq be the q-distance matrix of T.

Example

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Dq =       1 1 + q 1 + q 1 + q + q2 1 1 1 1 + q 1 + q 1 1 + q 1 + q + q2 1 + q 1 1 + q 1 1 + q + q2 1 + q 1 + q + q2 1      

Determinant of the q-distance matrix of a tree

Theorem

Let T be a tree with n vertices and let Dq be the q-distance matrix of T. Then detDq = (−1)n−1(n − 1)(1 + q)n−2.

q-Laplacian

Let Lq = I − qA + q2(Diag(δ) − I). If q = −1, then D−1

q

= − 1 1 + q Lq + a rank one matrix The determinant of Lq is related to the Ihara zeta function.

• A. Terras, Zeta functions of graphs, A stroll through the garden,

Cambridge Univ Press, 2011.

Exponential distance matrix

For a tree with n vertices, define Eq = [qd(i,j)] Dq and Eq are closely related: (1 − q)Dq = J − Eq. Theorem: (i) det Eq = (1 − q2)n−1. (ii) If q = ±1, then E −1

q

=

1 1−q2 Lq.

Squared distance matrix of a tree

Let T be a tree with n vertices and let D be the distance matrix of

• T. The squared distance matrix ∆ of T is defined as

∆ = D ◦ D = [d(i, j)2].

Consider the tree:

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∆ =           1 4 4 9 9 16 1 1 1 4 4 9 4 1 4 9 9 16 4 1 4 1 1 4 9 4 9 1 4 9 9 4 9 1 4 1 16 9 16 4 9 1          

Determinant of the squared distance matrix

Theorem Let T be a tree with n vertices and let ∆ be the squared distance matrix of T. Let d1, d2, . . . , dn be the degree sequence of

• T. Let τi = 2 − di, i = 1, . . . , n. Then

det ∆ = (−1)n4n−2   (2n − 1)

n

• i=1

τi − 2

n

• i=1
• j=i

τj    .

Theorem Let T be a tree and let ∆ be the squared distance matrix of T. Then ∆ is nonsingular if and only if T has at most

• ne vertex of degree 2.

Edge orientation matrix

Let T be a tree with V (T) = {1, . . . , n}. We assign an orientation to each edge of T. Definition The edge orientation matrix of T is the (n − 1) × (n − 1) matrix H defined as follows. The rows and the columns of H are indexed by the edges of T. The (e, f )-element of H is 1(−1) if the corresponding edges of T are similarly (oppositely) oriented. The diagonal elements of H are set to be 1.

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H =               1 1 −1 1 1 −1 1 −1 1 1 1 1 −1 −1 1 −1 1 −1 −1 1 1 −1 −1 1 −1 1 −1 1 −1 −1 1 −1 1 −1 1 −1 1 −1 −1 −1 1 1 −1 1 −1 −1 1 1 1 1 1 −1 1 −1 1 −1 −1 −1 −1 −1 1 1 −1 −1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1              

Determinant of the edge orientation matrix

Theorem

Let T be a directed tree with n vertices and let H be the edge

• rientation matrix of T. Then

det H = 2n−2

n

• i=1

τi.

Corollary

H is nonsingular if and only if no vertex in T has degree 2.

More identities

Recall: Q′DQ = −2I, LDL = −2L, Dτ = (n − 1)1 We have: Dδ = ∆τ, Q′∆Q = −2H If T has no vertex of degree 2, then H−1 = 1 2Q′     

1 τ1

· · ·

1 τ2

· · · . . . . . . ... . . . · · ·

1 τn

     Q

“Two-step” Laplacian of a tree

Let T be a tree with n vertices and with no vertex of degree 2. Let L be the Laplacian of T. The ”two-step” Laplacian of T is defined as L = 1 2L     

1 τ1

· · ·

1 τ2

· · · . . . . . . ... . . . · · ·

1 τn

     L. For i = j, the (i, j)-element of L is nonzero if and only if d(i, j) ≤ 2.

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L =             1 −1/2 −1/2 1 −4 1 3 −1/2 −1/2 −1/2 1 −1/2 −1/2 3 −1/2 −5 1 3 −1/2 −1/2 −1/2 1 −1/2 −1/2 3 −1/2 −4 1 1 −1/2 1 −1/2 −1/2 1 −1/2            

Inverse of the squared distance matrix of a tree

Theorem Let T be a tree with n vertices and with no vertex of degree 2. Let ∆ be the squared distance matrix of T and let L be the ”two-step” Laplacian of T. Then ∆−1 = −1 2L + a rank one matrix.

Inertia of the squared distance matrix

Theorem Let T be a tree with n vertices and let ∆ be the squared distance matrix of T. Let T have p pendant vertices and q vertices

• f degree 2. Then ∆ has

p negative eigenvalues and max{0, q − 1} zero eigenvalues.

Nullity of ∆ ≥ q − 1: Proof sketch

If vertex j has degree 2 and is adjacent to i, k, then Coli(∆) + Colk(∆) − 2Colj(∆) = 21 If dui = α, then duj = α + 1, duk = α + 2. Note that α2 + (α + 2)2 − 2(α + 1)2 = 2

Inertia of the edge orientation matrix

Theorem Let T be a directed tree with p pendant vertices and q vertices of degree 2. Let H be the edge orientation matrix of T. Then H has p − 1 positive eigenvalues and max{0, q} zero eigenvalues.

Conclusion

We discussed several instances of the formula D−1 = −1 2L + a rank one matrix where D is a “distance matrix” and L is a “Laplacian”. The squared distance matrix of a tree is an interesting matrix and it also fits in this framework.

Thank You!