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 Ht=exp⁡(−tL(G)) is the heat kernel (HK) of G, with t>0 denoting the time variable. For a vertex u∈V(G), the uth element of the diagonal of the HK is defined as Ht(u,u)=(exp⁡(−tL(G)))uu=∑k=0∞((−tL(G))k)uuk!, and HE(G)=∑i=1ne−tλi=∑u=1nHt(u,u) is the HK trace of G, where λ1,λ2,⋯,λn 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.