Discrete Integral and Discrete Derivative on Graphs and Switch Problem of Trees

For a vertex and edge weighted (VEW) graph G with weight function fG let Wα,β(G)=∑{u,v}⊆V(G)[αfG(u)×fG(v)+β(fG(u)+fG(v))]dG(u,v) where, α,β∈ℝ dG(u,v) denotes the distance, minimum sum of weights across all paths connecting u,v∈V(G). Assume T is VEW tree, e∈ E(T) fails. If we reconnect two components T−e new ϵ≠e such that, Wα,β(Tϵ\e=T−e+ϵ) minimum, then ϵ called best switch (BS) e w.r.t. Wα,β. We define three notions: convexity, discrete derivative, integral for graphs. As an application notions, solve some BS problems positively trees. example, assume n-vertex tree. Then, inputs w,α,β ∈ℝ+, return ϵ, Tϵ\e, Wα,β(Tϵ\e) worst average time O(logn) O(1) where Wα,β w.