A note on heat kernel of graphs

Consider a simple undirected connected graph G, with D(G) and A(G) representing its degree and adjacency matrices, respectively. Furthermore, L(G) = D(G) - A(G) is the Laplacian matrix of G, and H-t = exp (-tL(G)) is the heat kernel (HK) of G, with t > 0 denoting the time variable. For a vertex u is an element of V(G), the uth element of the diagonal of the HK is defined as H-t (u, u) = (exp(-tL(G)))uu = Sigma(infinity)(k=0)((-tL(G))k)(uu) = Sigma(infinity)(t=0) ((-tL(G))(k))(uu)/k!, and HE(G) = Sigma(n)(i=1) e(i)(-t lambda) = Sigma(n)(u=1) H-t(u, u) is the HK trace of G, where lambda(1), lambda(2), center dot center dot center dot , lambda(n) denote the eigenvalues of L(G). This study provides new computational formulas for the HK diagonal entries of graphs using an almost equitable partition and the Schur complement technique. We also provide bounds for the HK trace of the graphs.