Kernels in the Selberg trace formula on the k-regular tree and zeros of the Ihara zeta function

There are various kernels that we can associate to the k-regular tree, Tk . Here we will take a few of those kernels and insert them into the Selberg trace formula on Tk . Amongst the results, we find a simple proof for the Graph Prime Number Theorem and a proof of McKay's Theorem regarding the distribution of the adjacency matrix eigenvalues for a sequence of regular graphs of fixed degree whose number of vertices go to infinity. Using a simple change of variables, we are able to find the limiting distribution of the t-values of the Ihara zeta function for a (q + 1)-regular graph. We then provide evidence that, when our graphs are random, the level spacings of the t-values have the same distribution as the Gaussian Orthogonal Ensemble. And even when the graph is not random, the level spacings of the t-values are the same as the level spacings of the eigenvalues of the associated adjacency matrix.