Maximization of combinatorial Schrödinger operator’s smallest eigenvalue with Dirichlet boundary condition

For a nonnegative potential function qq and a given locally finite graph GG, we study the combinatorial Schrödinger operator Lq(G)=ΔG+qLq(G)=ΔG+q with Dirichlet boundary condition on a proper finite subset SS of the vertex set of GG such that the induced subgraph on SS is connected. Let Υp={q∈Lp(S):q(x)≥0,∑x∈Sqp(x)≤1}Υp={q∈Lp(S):q(x)≥0,∑x∈Sqp(x)≤1}, for 1≤p<∞1≤p<∞. We prove the existence and uniqueness of the maximizer of the smallest Dirichlet eigenvalue of Lq(G)Lq(G), whenever the potential function q∈Υpq∈Υp. Furthermore, we also establish the analogue of the Euler–Lagrange equation on graphs.