GREEN MATRICES OF WEIGHTED GRAPHS WITH PENDANT VERTICES. Midsummer Combinatorial Workshop XX Prague, 28 July- 1 August 2014

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1 GREEN MATRICES OF WEIGHTED GRAPHS WITH PENDANT VERTICES Silvia Gago joint work with Angeles Carmona, Andrés M. Encinas, Margarida Mitjana Midsummer Combinatorial Workshop XX Prague, 28 July- 1 August 2014 Department of Applied Mathematics 3 and 1, Universitat Politècnica de Catalunya, UPCTech SPAIN

2 OUTLINE 1 Discrete Potential Theory Definitions and concepts 2 The Poisson equation Applications Effective Resistances Kirchhoff index 3 Previous results 4 Adding new pendant vertices Adding one pendant vertex Adding several pendant vertices Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 2/28

3 DISCRETE POTENTIAL THEORY

4 DEFINITIONS AND CONCEPTS A weighted graph Γ = (V, E, c) is composed by: V is a set of elements called vertices. E is a set of elements called edges. c : V V [0, ) is an application named conductance associated to the edges. u, v are adjacent, u v iff c(u, v) = c uv 0. The degree of a vertex u is d u = v V c uv. x2 c23 x3 c12 c15 c15 x5 c35 c34 x4 x1 c56 c46 c17 c57 x6 x7 Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 4/28

5 DEFINITIONS AND CONCEPTS x2 c23 x3 c12 c15 c15 x5 c35 c34 x4 x1 c56 c46 c17 c57 x6 x7 Consider the set of real-valued functions on V, C(V ), where every function on V is a vector in R n. C(V ) R n u [u(x 1),..., u(x n)] T. Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 5/28

6 DEFINITIONS AND CONCEPTS x2 c23 x3 c12 c15 c15 x5 c35 c34 x4 x1 c56 c46 c17 c57 x6 x7 Consider the set of real-valued functions on V, C(V ), where every function on V is a vector in R n. C(V ) R n u [u(x 1),..., u(x n)] T. The scalar product is u, v = x V u(x)v(x) = uv T. Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 5/28

7 DEFINITIONS AND CONCEPTS c23 x3 ω3 ω2 x2 c12 c15 c35 c34 c15 x5 ω5 x4 ω4 ω1 x1 c56 c46 c17 c57 ω7 x7 x6 ω6 Consider the set of real-valued functions on V, C(V ), where every function on V is a vector in R n. C(V ) R n u [u(x 1),..., u(x n)] T. The scalar product is u, v = x V u(x)v(x) = uv T. A weight is a positive function ω : V (0, + ). The set of weights is denoted by Ω(V ). Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 5/28

8 DEFINITIONS AND CONCEPTS Any matrix K M n n is the kernel of the endomorphism K of C(V ) that assigns to any u C(V ) the function v. K : C(V ) C(V ) u K(u) = Ku = v. Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 6/28

9 DEFINITIONS AND CONCEPTS Any matrix K M n n is the kernel of the endomorphism K of C(V ) that assigns to any u C(V ) the function v. K : C(V ) C(V ) u K(u) = Ku = v. Conversely, each endomorphism K of C(V ) determines the kernel given by (K) x,y = K(ɛ y), ɛ x for any x, y V. Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 6/28

10 DEFINITIONS AND CONCEPTS Any matrix K M n n is the kernel of the endomorphism K of C(V ) that assigns to any u C(V ) the function v. K : C(V ) C(V ) u K(u) = Ku = v. Conversely, each endomorphism K of C(V ) determines the kernel given by (K) x,y = K(ɛ y), ɛ x for any x, y V. Common used operators Operator Kernel Identity I I Projector on σ along τ P σ,τ (u) = τ, u σ P σ,τ = σ τ Projector on ω P ω(u) = ω, u ω P ω = ω ω Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 6/28

11 LAPLACIAN OPERATOR AND LAPLACIAN MATRIX The Laplacian operator L is L : C(V ) C(V ) u L(u) = x,y V cxy[u(x) u(y)]. The corresponding Laplacian matrix is { d i if i = j, (L) ij = c ij if i j. For a simple graph, c ij = 1 for any i, j V and the applications of its spectrum have been widely studied in Spectral Graph Theory (The Laplacian Spectrum of Graphs, B. Mohar). The general case has not been so well studied. Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 7/28

12 DISCRETE POTENTIAL THEORY Definition The Schrödinger L q operator on a network with potential q is a generalization of the Laplacian operator L defined as L q : C(V ) C(V ) u L q(u) = L(u) + qu, where qu C(V ) is defined as qu(x) = q(x)u(x), for all x V. The Schrödinger matrix of a network Γ with potential Q is defined as L q = L + Q, Q = diag(q 1,..., q n). Observe that L q can be also considered as the Laplacian operator of a network with loops of value q i associated of each vertex. q3 q2 c23 x3 x2 c25 c12 q1 c15 x1 q5 x5 c35 c56 c34 x4 c46 q4 c57 x6 c17 x7 q6 Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 8/28 q7

13 DISCRETE POTENTIAL THEORY Any Schrödinger operator L q is selfadjoint. If L q is positive definite, then it is invertible and its inverse is called Green's operator. However we are interested in those Schrödinger operators L q who are positive semi-definite. Now consider q ω = 1 L(ω) C(V ), it is called the potential determined ω by a weight ω Ω(V ). Proposition The Schrödinger operator L q with potential q is positive semidefinite iff there exists a weight ω Ω(V ) and a real number λ 0 such that q = q ω + λ. In this case λ is the lowest eigenvalue and its associated eigenfunctions are multiples of ω. Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 9/28

14 DISCRETE POTENTIAL THEORY Any Schrödinger operator L q is selfadjoint. If L q is positive definite, then it is invertible and its inverse is called Green's operator. However we are interested in those Schrödinger operators L q who are positive semi-definite. Now consider q ω = 1 L(ω) C(V ), it is called the potential determined ω by a weight ω Ω(V ). Proposition The Schrödinger operator L q with potential q is positive semidefinite iff there exists a weight ω Ω(V ) and a real number λ 0 such that q = q ω + λ. In this case λ is the lowest eigenvalue and its associated eigenfunctions are multiples of ω. Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 9/28

15 THE POISSON EQUATION

16 THE POISSON EQUATION Our goal is to solve the Poisson equation for F on V with data f V, that is to find u V such that L q(u) = f. It L q is positive semidefinite, then the operator that assigns to each function f C(V ) the unique u C(V ) such that L q(u) = f P ω(f), is called orthogonal Green's operator G q, and its kernel G q is called Green's matrix. Notice that G q(ω) = { 1 λ if λ 0 0 if λ = 0. Some applications: by knowing the Green's matrix G q we can compute the effective resistances and the Kirchhoff index of the weighted graph. Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 11/28

17 APPLICATIONS: EFFECTIVE RESISTANCES Klein and Rándic (93) proposed a distance measure between two vertices of electrical networks called the effective resistance u v R 1 R 2 d(u, v) = 2 u R uv = 1 1 R R 2 v u v R 1 R 2 d(u, v) = 2 u v R 3 R 4 1 R uv = 1 R 1 +R R 3 +R 4 Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 12/28

18 APPLICATIONS: EFFECTIVE RESISTANCES Bendito, Carmona, Encinas, Gesto (2008) proved that the resistance distance between any pair of vertices x, y V can be computed as the solution of the following linear problem L(u) = π x,y, where L is the Laplacian operator of the weighted graph and π x,y = ε x ε y is called the dipole between x and y. Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 13/28

19 APPLICATIONS: EFFECTIVE RESISTANCES Bendito, Carmona, Encinas, Gesto (2008) proved that the resistance distance between any pair of vertices x, y V can be computed as the solution of the following linear problem L(u) = π x,y, where L is the Laplacian operator of the weighted graph and π x,y = ε x ε y is called the dipole between x and y. Proposition [Bendito, Carmona, Encinas, Gesto (2009)] For any x, y V it is verified that R λ,ω (x, y) = (Gq)x,x ω 2 x + (Gq)y,y ω 2 y 2(Gq)x,y ω xω y. Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 13/28

20 APPLICATIONS: KIRCHHOFF INDEX Introduced in Chemistry as a better alternative to other parameters for discriminating among different molecules with similar shapes and structures. The Kirchhoff index or Total Resistance of the weighted graph is the sum of all the effective resistances between any pair of vertices of the weighted graph. Corollary [Bendito, Carmona, Encinas, Gesto (2009)] For any x, y V it is verified that k(λ, ω) = trace(g q) { 1 λ if λ 0 0 if λ = 0. Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 14/28

21 PREVIOUS RESULTS

22 PREVIOUS RESULTS The Green matrix has been computed for Cluster graphs Some composite weighted graphs Generalized linear polyominoes Join graphs Perturbations of an edge Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 16/28

23 PERTURBING THE CONDUCTANCE OF AN EDGE Corollary [Carmona, Encinas, Mitjana] Consider σ C(V ), the perturbation of F by σ and the operator H σ = F + P σ then when σ, ω = 0 whereas, when σ, ω = 0 H 1 = G 1 β G H = G G(σ), σ P G(σ), [λp G(σ) σ, ω ( P G(σ),ω P ω,g(σ) ) (1 + σ, ω ) Pω ], where β = λ(1 + G(σ), σ ) + σ, ω 2. Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 17/28

24 ADDING PENDANT VERTICES Goal To study the perturbation of the Green matrix of a weighted graph caused by the perturbation due to add a pendant vertex, relating both Green matrices. Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 18/28

25 ADDING NEW PENDANT VERTICES

26 ADDING ONE PENDANT VERTEX X New network: Γ = (V, E, c ), V = V {x }, E = E {e x,x }, c = c (x, x ) > 0 New weight: ω x Ω(V ), such that ω x i = ω xi / 1 + ωx 2 for any x i V, i = 0,..., n x c x It holds ω i /ω j = ω i/ω j for any i, j V. Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 20/28

27 ADDING ONE PENDANT VERTEX X The Schrödinger operator of Γ is L q(u) = L (u) + q u, with q = q ω + λ and q ω = 1 ω L (ω ) C(V ). The relation between the kernels of both Schrödinger operators is given by ( L L q = q c ) ε x c ε x q x,x, where L q = L q + c ω x ω x P εx is the kernel of the perturbed operator L q = L q + P σx,x, with σ x,x = c ω x ω x ε x and q x,x = ωx c ω + λ R. x Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 21/28

28 ADDING ONE PENDANT VERTEX Proposition The kernel of L 1 q is (L q ) 1 = 1 c H H 1 ε q x x,x c q x,x ε xh 1 1 q x,x c (q x,x )2 ε xh 1 ε x, (1) where H 1 is the kernel of the operator H 1 = G λ,ω 1 [ ( ) λp Gλ,ω (σ β x,x ) σ x,x, ω P Gλ,ω (σ x,x ),ω P ω,gλ,ω (σ x,x ) and β = λ(1 + G λ,ω (σ x,x ), σ x,x ) + σ x,x, ω 2. (1 + σ x,x, ω ) P ω ], Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 22/28

29 ADDING SEVERAL PENDANT VERTICES X 1,..., X M New network: Γ = (V, E, c ), V = V {x 1,..., x m}, E = E {e x1,x,..., e 1 x m,x } and c m i = c (x i, x i ) > 0 for i = 1,..., m New weight: ω x Ω(V ), such that ω x = ω x / any x i V, i = 0,..., n 1 + m i=1 ω2 x i for x 3 c 3 x 2 c 2 x 2 c 1 x 1 x m x 1 c m x m Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 23/28

30 ADDING SEVERAL PENDANT VERTICES X 1,..., X M The Schrödinger operator of Γ is L q(u) = L (u) + q u, with q = q ω + λ and q ω = 1 ω L (ω ). The relation between the kernels of both Schrödinger operators is given by ( ) L L q = q B B, D q ω x i where L q = L q + m i=1 c i ω xi P εxi, B = ( c x 1 ε x1,..., c x m ε xm ), and D q is an invertible diagonal matrix having in the diagonal the potential of the new vertices q x i,x = c ω xi i i ω + λ 0. x i Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 24/28

31 ADDING SEVERAL PENDANT VERTICES Proposition The kernel of (L q ) 1 is ( (L q ) 1 = where H 1 is the kernel of the operator H 1 = G + αp ω + H 1 H 1 BD 1 ) D 1 B H 1 D 1 + D 1 B H 1 BD 1, m β i [P G(σi ),ω P ω,g(σi )] i=1 m γ ij P G(σi ),G(σ j ), i,j=1 1 and (b ij ) = (I + G(σ j ), σ i ) 1, α = λ + b r,s G(σ j ), σ i σ s, ω, r,s ( m m ) ( m ) β i = α b ir σ r, ω and γ ij = b ij α b ir σ r, ω b jr σ r, ω. r=1 r=1 r=1 Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 25/28

32 EXAMPLE Green matrix of a Star graph S n We compute the Green matrix of a Star graph S n with constant weight ω = 1 n j by adding n 2 pendant vertices to a S 2. ( 1 + λ 1 Starting from L λ,2 = λ ) ( ) jn 2, B =, 0 n 2 ( λ 2 +nλ+1 H = λ+1 1 G λ,n = n 1 n(λ+n) 1 n(λ+n). 1 n(λ+n) ) 1 H 1 = 1 + λ 1 n(λ+n) (n 1)λ+(n 2 n 1) n(λ+1)(λ+n). λ+n+1 n(λ+n)(λ+1) 1+λ λ 2 +nλ 1 λ 2 +nλ 1 λ 2 +nλ λ 2. +nλ+1 (λ+1)(λ 2 +nλ) 1... n(λ+n)... λ+n+1 n(λ+n)(λ+1) (n 1)λ+(n 2 n 1) n(λ+1)(λ+n) Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 26/28

33 TO KNOW MORE ABOUT THIS TOPIC... Advanced Course on Combinatorial Matrix Theory Date: February 2015 Location: Centre de Recerca Matemàtica (CRM), Barcelona Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 27/28

34 Thanks for your attention Děkuji za pozornost Gracias por su atención Green Matrices of Weighted Graphs with Pendant Vertices MCW XX, Prague 28 July-2 August 28/28