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The best constant of three kinds of the discrete Sobolev inequalities on the complete graph. (English) Zbl 1310.46037
Summary: We introduce a discrete Laplacian \(A\) on the complete graph with \(N\) vertices, that is, \(K_N\). We obtain the best constants of three kinds of discrete Sobolev inequalities on \(K_N\). The background of the first inequality is the discrete heat operator \((d/dt + A + a_0 I) \cdots (d/dt + A + a_{M-1} I)\) with positive distinct characteristic roots \(a_{0}, \dots, a_{M-1}\). The second one is the difference operator \((A + a_{0} I) \cdots (A + a_{M-1} I)\) and the third one is the discrete polyharmonic operator \(A^M\). Here \(A\) is an \(N \times N\) real symmetric positive-semidefinite matrix whose eigenvector corresponding to zero eigenvalue is \(\mathbf 1 = ^t(1, 1,\dots, 1)\). A discrete heat kernel, a Green’s matrix and a pseudo Green’s matrix are obtained by means of \(A\).

MSC:
46E39 Sobolev (and similar kinds of) spaces of functions of discrete variables
35K08 Heat kernel
35R02 PDEs on graphs and networks (ramified or polygonal spaces)
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