A hierarchical structure for the sharp constants of discrete Sobolev inequalities on a weighted complete graph
Summary: This paper clarifies the hierarchical structure of the sharp constants for the discrete Sobolev inequality on a weighted complete graph. To this end, we introduce a generalized-graph Laplacian \(\boldsymbol{A} = \boldsymbol{I} - \boldsymbol{B}\) on the graph, and investigate two types of discrete Sobolev inequalities. The sharp constants \(C_0(N; a)\) and \(C_0(N)\) were calculated through the Green matrix \(\boldsymbol{G}(a) = (\boldsymbol{A} + a \boldsymbol{I})^{- 1}(0 < a < \infty)\) and the pseudo-Green matrix \(\boldsymbol{G}_\ast = \boldsymbol{A}^\dagger\). The sharp constants are expressed in terms of the expansion coefficients of the characteristic polynomial of \(\boldsymbol{A}\). Based on this new discovery, we provide the first proof that each set of the sharp constants \(\{C_0(n; a) \}_{n = 2}^N\) and \(\{C_0(n) \}_{n = 2}^N\) satisfies a certain hierarchical structure.
scientific article; zbMATH DE number 6893757