Abstract: | Let Γ be a graph endowed with a reversible Markov kernel p, whose associated operator P is defined by (Pf(x) = {sum }_{y} p(x, y)f(y)). We assume that the kernels pn(x, y) associated to Pn satisfy Gaussian upper bounds but do not assume they satisfy the Hölder continuity property and the temporal regularity. Denote by L = I ? P the discrete Laplacian on Γ. This article shows the weighted weak type (1, 1) estimates and the weighted Lp norm inequalities for the spectral multipliers of L. We also obtain the weighted Lp norm inequalities for the commutators of the spectral multipliers of L with BMO functions which are new even for the unweighted case. |