Group inverse matrix of the normalized laplacian on subdivision graphs  ´ Angeles Carmona, Margarida Mitjana, Enric Mons´ o Departament Matem`atiques, Universitat Polit` ecnica de Catalunya. E-mail: enrique.monso@upc.edu Abstract. In this paper, given a network we consider a subdivision network of it and we show how the group inverse matrix of the normalized laplacian on the subdivision network is related to the group inverse matrix of the normalized laplacian of the initial given network. Our approach establishes a relationship between solutions of related Poisson problems on both structures and takes advantage on the properties of the group inverse matrix. Keywords. Subdivision network, normalized laplacian, group inverse, Poisson problem 1 Preliminaries In this paper Γ = (V,E,c) denotes a simple network; that is, a finite, with no loops, nor multiple edges, connected graph, with n vertices in V and m edges in E. We call conductance to the symmetric function c : V × V → [0, +∞) satisfying c(x, y) > 0 iff x ∼ y, which means that {x, y}∈ E. For every vertex in V, let k(x)= ∑ y∈V c(x, y) be the degree of vertex x, then vol(Γ) = ∑ x∈V k(x). Let C (V ) be the set of real functions on V, the normalized laplacian of Γ, introduced by Chung and Langlands in [2], is the linear operator L : C (V ) →C (V ) that assigns to each u ∈C (V ) and at each x ∈ V L(u)(x)= 1  k(x)  y∈V c(x, y)  u(x)  k(x) − u(y)  k(y)  . Easily ker(L)= span( √ k) and given f ∈C (V ) the Poisson problem, i.e. the linear system L(u)(x)= f (x), is compatible iff f ⊥ √ k. In this case, two different solutions differ up to a multiple of √ k, so there exists a unique solution orthogonal to √ k to every compatible linear system L(u)= f (Fredholm’s alternative).  This work has been partially supported by the spanish Programa Estatal de I+D+i del Ministerio de Econom´ ıa y Competitividad, under the project MTM2014-60450-R