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 C0(a) and C0 are calculated using the Green matrix G(a)=(A+aI)−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+aI and A except an eigenvalue 0.