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

作者：Takemura Kazuo^*; Kametaka Yoshinori; Nagai Atsushi

来源：Symmetry-Basel, 2018, 10(1): 1.

DOI：10.3390/sym10010001

• Sobolev inequality
• sharp constant
• complete graph
• graph Laplacian
• Green matrix

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 + aI)(-1) (0 < a < infinity) 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 and {C-0 (n)(n=2)(N)satisfies a certain hierarchical structure.

• 出版日期2018-1