A Hierarchical Structure for the Sharp Constants of Discrete Sobolev Inequalities on a Weighted Complete Graph

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 A = I − 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 G ( a ) = ( A + a I ) − 1 ( 0 < a < ∞ ) and the pseudo-Green matrix G ∗ = A † . The sharp constants are expressed in terms of the expansion coefficients of the characteristic polynomial of 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.