Two types of discrete Sobolev inequalities on a weighted Toeplitz graph

Abstract In this paper, two types of discrete Sobolev inequalities that correspond to the generalized graph Laplacian A on a weighted Toeplitz graph are obtained. The sharp constants C 0 ( a ) and C 0 are calculated using the Green matrix G ( a ) = ( A + a I ) − 1 ( 0 a ∞ ) and pseudo-Green matrix G ⁎ = A † (Penrose–Moore generalized inverse matrix of A ). The sharp constants are expressed as reciprocals of the harmonic mean corresponding to eigenvalues of each matrix A + a I and A except an eigenvalue 0.