On the Lichnerowicz estimate for Laplacians on graphs under spectral curvature assumptions
A triple \(G=(V,w,m)\) that consists of a finite vertex set \(V\), a symmetric function \(w: V\times V\rightarrow [0,\infty)\) which is zero on diagonal, and a vertex measure \(m: V\rightarrow (0,\infty)\) is called a finite weighted and measured graph. Let \(C(V)\) denote the set of real valued functions defined on \(V\). On the graph \(G\), the discrete Laplacian \(\Delta: C(V)\rightarrow C(V)\) is given in the following way \[ \Delta f(x):=\sum_{y\in V} \frac{w(x,y)}{m(x)} (f(y)-f(x)),\quad x\in V. \] Let \(K>0\), \(n\in(0,\infty]\) and \(\rho\in C(V)\). It is said that the \((K,n,\rho)\)-spectrally positive Ricci curvature dimension condition on \(G\) is satisfied if and only if the following two conditions are satisfied: \par i) the variable curvature dimension condition \(CD(\rho,n)\) holds \[ \Gamma_2(f)(x)\geq \rho(x)\Gamma(f)(x)+\frac{1}{n}(\Delta f)^2(x),\quad f\in C(V), \quad x\in V. \] ii) as well as \[ -\frac{1}{2}\langle -\Delta f,f\rangle+\langle \rho f,f\rangle\geq K\langle f,f\rangle,\quad f\in C(V). \] The following eigenvalue estimate is established and proved. Let \(K>0\), \(n\in(1,\infty]\) and \(\rho\in C(V)\). Let \(G=(V,w,m)\) be a finite weighted and measured graph on which the \((K,n,\rho)\)-spectrally positive Ricci curvature dimension condition is satisfied. Then, \[ \lambda\geq \frac{n}{n-1}K, \] where \(\lambda=\lambda(G)\) is the first positive eigenvalue of \(\Delta\) in \(\ell^2(V,m)\).
scientific article; zbMATH DE number 7606794