Connection heat kernel on connection lattices and connection discrete torus

By a connection graph we mean an underlying graph with a connection which associates edge set with an orthogonal group. This paper centers its investigation on the connection heat kernels on connection lattices and connection discrete torus. For one dimensional connection lattice, we derive the connection heat kernel expression by doing the Taylor expansion on the exponential function involving normalized connection Laplacian. We introduce a novel connection called product connection and prove that the connection heat kernel on arbitrary high dimensional lattice with product connection equals the Kronecker sum of one dimensional connection lattices' connection heat kernels. Furthermore, if the connection graph is consistent, we substantiate the interrelation between its connection heat kernel and its underlying graph's heat kernel. We define a connection called quotient connection such that discrete torus with quotient connection can be considered as a quotient graph of connection lattice, whose connection heat kernel is demonstrated to be the sum of connection lattices' connection heat kernels. In addition, we derive an alternative expression of connection heat kernel on discrete torus whenever its quotient connection is a constant connection, yielding an equation as a connection graph's trace formula.