An Introduction to Resolvent kernels for Dirichlet BVPs on Finite Networks

Throughout the paper, Γ = (V,E) denotes a simple, finite and connected graph without loops, with vertex set V and edge set E. Two different vertices, x, y ∈ V , are called adjacent, which will be represented by x ∼ y, if {x, y} ∈ E. Given a vertex subset F ⊂ V , we denote by F c its complementary in V and we call boundary and closure of F , the sets δ(F ) = {x ∈ V : x ∼ y for some y ∈ F} and F̄ = F ∪ δ(F ), respectively. If F ⊂ V is a proper subset, we say that F is connected if for any x, y ∈ V there exists a path joined x and y whose vertices are all in F . It is easy to prove that F̄ is connected when F is. The sets of functions and non-negative functions on V are denoted by C(V ) and C(V ) respectively. If u ∈ C(V ), its support is given by supp(u) = {x ∈ V : u(x) 6= 0}. Moreover, if x ∈ V , we denote by εy the Dirac function; that is, εy(x) = 0 if x 6= y and εy(y) = 1. If F is a non empty subset of V , its characteristic function is denoted by χ F and we can consider the sets C(F ) = {u ∈ C(V ) : supp(u) ⊂ F} and C(F ) = C(F ) ∩ C(V ). We call weight on F any function σ ∈ C(F ) such that supp(σ) = F . The set of weights on F is denoted by Ω(F ). We call conductance on Γ a function c : V ×V −→ IR such that c(x, y) > 0 iff x ∼ y. We call weighted network any triple (Γ, c, ν), where c is a conductance on Γ and ν ∈ Ω(V ). In what follows we consider fixed the network (Γ, c, ν) and we refer to it simply by Γ. The function κ ∈ C(V ) defined as κ(x) = ∫