The Kirchhoff Index of Cluster Networks uz 1 , E. Bendito 1 , A. - PowerPoint PPT Presentation

The Kirchhoff Index of Cluster Networks uz 1 , E. Bendito 1 , A. Carmona 1 and A.M. Encinas 1 C. Ara´ Budapest, Eurocomb 2011 1 Dept. Matem` atica Aplicada III Universitat Polit` ecnica de Catalunya, Barcelona

Kirchhoff Index in Chemistry � The Kirchhoff Index was introduced in Chemistry Ara´ uz, Bendito, Carmona, Encinas (UPC) Matem` atica Aplicada III dept. 2 / 16 The Kirchhoff Index of Cluster Networks

Kirchhoff Index in Chemistry � The Kirchhoff Index was introduced in Chemistry � Kirchhoff Index of a molecule: � 2 � atomic displacements � from equilibrium positions atoms Ara´ uz, Bendito, Carmona, Encinas (UPC) Matem` atica Aplicada III dept. 2 / 16 The Kirchhoff Index of Cluster Networks

Kirchhoff Index in Chemistry � The Kirchhoff Index was introduced in Chemistry � Kirchhoff Index of a molecule: � 2 � atomic displacements � from equilibrium positions atoms small Kirchhoff Index ⇒ the atoms are very rigid in the molecule H H C H H H H C H C C C H H H H C H H C H H H Ara´ uz, Bendito, Carmona, Encinas (UPC) Matem` atica Aplicada III dept. 2 / 16 The Kirchhoff Index of Cluster Networks

Kirchhoff Index in Mathematics � In the mathematic field, it is interesting to find possible relations between the Kirchhoff Indexes of composite networks and those of their factors k 2 k 1 k k 3 k 4 Ara´ uz, Bendito, Carmona, Encinas (UPC) Matem` atica Aplicada III dept. 3 / 16 The Kirchhoff Index of Cluster Networks

Our objective � We have worked with a generalization of the classical Kirchhoff Index k k ( ω ) ��� Ara´ uz, Bendito, Carmona, Encinas (UPC) Matem` atica Aplicada III dept. 4 / 16 The Kirchhoff Index of Cluster Networks

Our objective � We have worked with a generalization of the classical Kirchhoff Index k k ( ω ) ��� � We have worked with on a particular family of composite networks: a generalized notion of cluster networks same different! same base base same same same Ara´ uz, Bendito, Carmona, Encinas (UPC) Matem` atica Aplicada III dept. 4 / 16 The Kirchhoff Index of Cluster Networks

Our objective � We have worked with a generalization of the classical Kirchhoff Index k k ( ω ) ��� � We have worked with on a particular family of composite networks: a generalized notion of cluster networks same different! same base base same same same � Our objective is to determine the Kirchhoff Index of generalized cluster networks in terms of the ones of their factors. Ara´ uz, Bendito, Carmona, Encinas (UPC) Matem` atica Aplicada III dept. 4 / 16 The Kirchhoff Index of Cluster Networks

Notations and basic results � Γ = ( V, E, c ) finite connected network ∗ V = { x 1 , . . . , x n } ∗ { x i , x j } ∈ E has conductance c ij > 0 Ara´ uz, Bendito, Carmona, Encinas (UPC) Matem` atica Aplicada III dept. 5 / 16 The Kirchhoff Index of Cluster Networks

Notations and basic results � Γ = ( V, E, c ) finite connected network ∗ V = { x 1 , . . . , x n } ∗ { x i , x j } ∈ E has conductance c ij > 0 n � ω ∈ R n , ω 2 ω i > 0 , � i = 1 weight on V i =1 Ara´ uz, Bendito, Carmona, Encinas (UPC) Matem` atica Aplicada III dept. 5 / 16 The Kirchhoff Index of Cluster Networks

Notations and basic results � Γ = ( V, E, c ) finite connected network ∗ V = { x 1 , . . . , x n } ∗ { x i , x j } ∈ E has conductance c ij > 0 n � ω ∈ R n , ω 2 ω i > 0 , � i = 1 weight on V i =1 standard inner product on R n � �· , ·� n u , v ∈ R n where � u , v � = � u i v i for i =1 Ara´ uz, Bendito, Carmona, Encinas (UPC) Matem` atica Aplicada III dept. 5 / 16 The Kirchhoff Index of Cluster Networks

Notations and basic results � Γ = ( V, E, c ) finite connected network ∗ V = { x 1 , . . . , x n } ∗ { x i , x j } ∈ E has conductance c ij > 0 n � ω ∈ R n , ω 2 ω i > 0 , � i = 1 weight on V i =1 standard inner product on R n � �· , ·� n u , v ∈ R n where � u , v � = � u i v i for i =1 � L = ( L ij ) n combinatorial Laplacian i , j =1 n where L ii = � c ik and L ij = − c ij for i � = j k =1 Ara´ uz, Bendito, Carmona, Encinas (UPC) Matem` atica Aplicada III dept. 5 / 16 The Kirchhoff Index of Cluster Networks

Schr¨ odinger matrix � L q = L + Q Schr¨ odinger matrix with potential q ∗ q ∈ R n potential on Γ ∗ Q = diag ( q 1 , . . . , q n ) diagonal matrix given by the values of q on V Ara´ uz, Bendito, Carmona, Encinas (UPC) Matem` atica Aplicada III dept. 6 / 16 The Kirchhoff Index of Cluster Networks

Schr¨ odinger matrix � L q = L + Q Schr¨ odinger matrix with potential q ∗ q ∈ R n potential on Γ ∗ Q = diag ( q 1 , . . . , q n ) diagonal matrix given by the values of q on V Remark But we want L q to be positive semi-definite! Ara´ uz, Bendito, Carmona, Encinas (UPC) Matem` atica Aplicada III dept. 6 / 16 The Kirchhoff Index of Cluster Networks

Schr¨ odinger matrix � L q = L + Q Schr¨ odinger matrix with potential q ∗ q ∈ R n potential on Γ ∗ Q = diag ( q 1 , . . . , q n ) diagonal matrix given by the values of q on V Remark But we want L q to be positive semi-definite! � We take q = q ω ∈ R n potential determined by ω , where ( q ω ) i = − ( ω ) − 1 ( L · ω ) i i Ara´ uz, Bendito, Carmona, Encinas (UPC) Matem` atica Aplicada III dept. 6 / 16 The Kirchhoff Index of Cluster Networks

Schr¨ odinger matrix � L q = L + Q Schr¨ odinger matrix with potential q ∗ q ∈ R n potential on Γ ∗ Q = diag ( q 1 , . . . , q n ) diagonal matrix given by the values of q on V Remark But we want L q to be positive semi-definite! � We take q = q ω ∈ R n potential determined by ω , where ( q ω ) i = − ( ω ) − 1 ( L · ω ) i i Results L q ω is positive semi-definite and singular � Ara´ uz, Bendito, Carmona, Encinas (UPC) Matem` atica Aplicada III dept. 6 / 16 The Kirchhoff Index of Cluster Networks

Schr¨ odinger matrix � L q = L + Q Schr¨ odinger matrix with potential q ∗ q ∈ R n potential on Γ ∗ Q = diag ( q 1 , . . . , q n ) diagonal matrix given by the values of q on V Remark But we want L q to be positive semi-definite! � We take q = q ω ∈ R n potential determined by ω , where ( q ω ) i = − ( ω ) − 1 ( L · ω ) i i Results L q ω is positive semi-definite and singular � L q ω · v = 0 iff v = aω with a ∈ R � Ara´ uz, Bendito, Carmona, Encinas (UPC) Matem` atica Aplicada III dept. 6 / 16 The Kirchhoff Index of Cluster Networks

Green matrix � L q ω · u = g Poisson equation Ara´ uz, Bendito, Carmona, Encinas (UPC) Matem` atica Aplicada III dept. 7 / 16 The Kirchhoff Index of Cluster Networks

Green matrix � L q ω · u = g Poisson equation L † the unique u ∈ R n such that � L † q ω f ∈ R n Green matrix − − → q ω ∗ L q ω · u = f − � f , ω � · ω ∗ � u , ω � = 0 Remark The Green matrix L † q ω is the Moore-Penrose inverse of the Schr¨ odinger matrix L q ω . Ara´ uz, Bendito, Carmona, Encinas (UPC) Matem` atica Aplicada III dept. 7 / 16 The Kirchhoff Index of Cluster Networks

Green matrix � L q ω · u = g Poisson equation L † the unique u ∈ R n such that � L † q ω f ∈ R n Green matrix − − → q ω ∗ L q ω · u = f − � f , ω � · ω ∗ � u , ω � = 0 Remark The Green matrix L † q ω is the Moore-Penrose inverse of the Schr¨ odinger matrix L q ω . Result L † � q ω · ω = 0 Ara´ uz, Bendito, Carmona, Encinas (UPC) Matem` atica Aplicada III dept. 7 / 16 The Kirchhoff Index of Cluster Networks

Effective resistances and Kirchhoff Index effective resistance between a pair of vertices � R ω − u j R ω ( x i , x j ) = u ⊤ · L q ω · u = u i ω i ω j where L q ω · u = e i − e j ω i ω j Ara´ uz, Bendito, Carmona, Encinas (UPC) Matem` atica Aplicada III dept. 8 / 16 The Kirchhoff Index of Cluster Networks

Effective resistances and Kirchhoff Index effective resistance between a pair of vertices � R ω − u j R ω ( x i , x j ) = u ⊤ · L q ω · u = u i ω i ω j where L q ω · u = e i − e j ω i ω j � r ω total resistance on a vertex r ω ( x i ) = v ⊤ · L q ω · v = v i − 1 n � v , ω � ω i where L q ω · v = e i − ω ω i Ara´ uz, Bendito, Carmona, Encinas (UPC) Matem` atica Aplicada III dept. 8 / 16 The Kirchhoff Index of Cluster Networks

Effective resistances and Kirchhoff Index effective resistance between a pair of vertices � R ω − u j R ω ( x i , x j ) = u ⊤ · L q ω · u = u i ω i ω j where L q ω · u = e i − e j ω i ω j � r ω total resistance on a vertex r ω ( x i ) = v ⊤ · L q ω · v = v i − 1 n � v , ω � ω i where L q ω · v = e i − ω ω i � k ( ω ) Kirchhoff index of Γ respect to ω n k ( ω ) = 1 � R ω ( x i , x j ) ω 2 i ω 2 j 2 i,j =1 Ara´ uz, Bendito, Carmona, Encinas (UPC) Matem` atica Aplicada III dept. 8 / 16 The Kirchhoff Index of Cluster Networks

Effective resistances and Kirchhoff Index Results r ω > 0 � Ara´ uz, Bendito, Carmona, Encinas (UPC) Matem` atica Aplicada III dept. 9 / 16 The Kirchhoff Index of Cluster Networks

Effective resistances and Kirchhoff Index Results r ω > 0 � � R ω is a distance on Γ Ara´ uz, Bendito, Carmona, Encinas (UPC) Matem` atica Aplicada III dept. 9 / 16 The Kirchhoff Index of Cluster Networks