Coverings, heat kernels and spanning trees (Q1277788)

This paper begins by defining a covering of a graph and also defines its Laplacian, eigenvalues and the heat kernel. Relations between the eigenvalues of a graph and the eigenvalues of its covering have been considered. In another section the heat kernel of an infinite \(k\)-regular tree has been derived, and heat kernels of some \(k\)-regular graphs have been discussed. Further, relations between the trace of the heat kernel and the number of spanning trees in a graph have also been considered. The paper also focuses on an old problem of determining the maximum number of spanning trees in a \(k\)-regular graph. Considering the zeta function of a graph, the upper and lower bounds for the maximum number of spanning trees in a \(k\)-regular graph on \(n\) vertices have been improved.