The heat kernel on a finite graph in different time-scales

Let G be a finite, weighted graph, and let T be a time-scale with a fixed point t 0 such that sup T = infinity. . In this paper, we construct the heat kernel on G in time-scale T in terms of a certain convolution series involving the heat operator acting on a parametrix, which is a fairly general function depending on the vertex set of G and the time variable t is an element of T. We develop some applications by choosing different parametrices and various time-scales. The results we obtain here extend, in part, aspects of the recent articles in that the time-scale considered in this paper is arbitrary.