  By the connection graph we mean an underlying weighted 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.
