College of Aerospace and Civil Engineering, Harbin Engineering University, Harbin 150001, China; College of Artificial Intelligence, Tianjin University of Science and Technology, Tianjin 300457, China; Key Laboratory of Dual Dielectric Power Technology, Hebei Hanguang Industry Co. Ltd., Handan 056017, China
Wei Ke
Key Laboratory of Dual Dielectric Power Technology, Hebei Hanguang Industry Co. Ltd., Handan 056017, China
Zhe Wang
Key Laboratory of Dual Dielectric Power Technology, Hebei Hanguang Industry Co. Ltd., Handan 056017, China
Haiyan Qiao
School of Information and Electrical Engineering, Hebei University of Engineering, Handan 056038, China; Corresponding author.
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.
