Green’s function in partial subdivision networks Ángeles Carmona, Margarida Mitjana, Enric Monsó Departament de Matemàtiques Universitat Politècnica de Catalunya. Spain Abstract In the present work, we define a partial subdivision network Γ S of a given network Γ, by inserting a new vertex in some selected edges of Γ, so that each of these edges is replaced by two new edges with conductances that fulfill the Kirchhoff series law on the new network. Then, we obtain an expression for the Green kernel of the partial subdivision network in terms of the Green kernel of the base network. For that, we show the relation between Poisson problems on the partial subdivision network and Poisson problems on the base network. Moreover, we also obtain the effective resistance and the Kirchhoff index of the partial subdivision network in terms of the corresponding parameters on the base network. Finally, as an example, we carry out the computations in the case of a star network in which we have subdivided the even edges. Keywords: Resistance distance, Green kernel, Kirchhoff Index, partial subdivision network MSC: 31C20, 15A09, 34B45 1 Introduction The subdivision graph of a given graph is obtained by dividing each edge into two edges by inserting one new node. This operation is sometimes called barycentric subdivision of the graph and is important when studying homeomorphic graphs; see [12]. In the literature, we can find some works studying parameters such as effective resistances, Kirchhoff Index, or spectra of subdivision graphs; see [4, 7, 9, 11, 13, 15]. In [8], balanced subdivision graph was studied in order to optimize the largest eigenvalue of the adjacency matrix. The subdivision of graphs is also used to construct equiarboreal graphs in [16]. The authors in [6], carried out the study of subdivision of networks. Specifically, we obtained the expression of the group inverse of the Laplacian matrix of the subdivision network in terms of the group inverse of the Laplacian matrix of the initial network. In the present paper, we first introduce the concept of partial subdivision of a network, that generalizes the usual subdivision network, in the sense that in the case of partial subdivision networks we only subdivide some selected edges. Moreover, in this work we obtain the expressions for the group inverse, the effective resistances and the Kirchhoff Index associated with singular Shcrödinger operators on the partial subdivided network as a function of the corresponding parameters in the base network. Our approach consists in interpreting a network as an electric circuit, and hence each selected edge has got assigned a positive number that corresponds with the conductance of a wire connecting two nodes. In addition, we also consider a weight in the set of vertices, that reflects the relevance of each node. When we perform the subdivision operation we interpret that 1