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The best constant of three kinds of discrete Sobolev inequalities on regular polyhedron. (English) Zbl 1288.46028
Summary: We consider three kinds of discrete Sobolev inequalities corresponding to a graph Laplacian \(\boldsymbol{A}\) on regular \(M\)-hedrons for \(M=4,6,8,12,20\). The discrete heat kernel \(\boldsymbol{H}(t)=\exp(-t\boldsymbol{A})\), the Green matrix \(\boldsymbol{G}(a)=(\boldsymbol{A}+a\boldsymbol{I})^{-1}\) and the pseudo Green matrix \(\boldsymbol{G}_*\) are obtained and investigated in a detailed manner. The best constants of the inequalities are given by means of eigenvalues of \(\boldsymbol{A}\).

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