A note on the graph’s resolvent and the multifilar structure

We consider the recently discovered [V. Ejov, J.A. Filar, S.K. Lucas, P. Zograf, Clustering of spectra and fractals of regular graphs, J. Math. Anal. Appl. 333 (2007) 236–246] threadlike structure of the plot representing d-regular graphs in the mean-variance coordinates of exponential sums of the graph spectra. In this note we demonstrate that this self-similar phenomenon is more ubiquitous by exhibiting it with the help of a different generating function, namely the mean and the variance of the resolvent of the adjacency matrix of the graph. We also discuss the location of non-Hamiltonian graphs within this geometric structure.