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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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