Summary: Let \(G\) be a finite, weighted graph, and let \(\mathbb{T}\) be a Time-scale with a fixed point \(t_0\) such that \(\sup\mathbb{T} = \infty\). In this paper we construct the heat kernel on \(G\) in Time-scale \(\mathbb{T}\) in terms of a certain convolution series involving he heat operator acting on a parametrix, which is a fairly general function depending on the vertex set of \(G\) and the time variable \(t\in\mathbb{T}\). We develop some applications by choosing different parametrices and various Time-scales. The results we obtain here do extend, in part, aspects of the recent articles in that the Time-scale considered in this paper is arbitrary.