For a graph G, let f_{ij} be the number of spanning rooted forests in which vertex j belongs to a tree rooted at i. In this paper, we show that for a path graph, f_{ij}'s can be expressed as products of Fibonacci numbers; for a cycle graph, they are products of Fibonacci numbers and Lucas numbers. The {\em doubly stochastic graph matrix} is the matrix F=(f_{ij})/f, where f is the total number of spanning rooted forests of G and n is the number of vertices in G. By the matrix forest theorem, F^{-1}=I+L, where L is the Laplacian matrix of G. Matrix F provides a proximity measure for graph vertices. We show that for the path graphs and the so-called T-caterpillars, some diagonal entries of F (which measure the self-connectivity of vertices) converge to \phi^{-1} or to 1-\phi^{-1}, where \phi is the golden ratio, as the number of vertices goes to infinity. Thereby, in the asymptotic, the corresponding vertices can be metaphorically considered as "golden introverts" and "golden extroverts," respectively. This metaphor is reinforced by a Markov chain interpretation of the doubly stochastic graph matrix, according to which F equals the overall transition matrix of a random walk with a random number of steps on G.