  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,
the f_{ij}'s can be expressed as the products of Fibonacci numbers; for a
cycle, they are products of Fibonacci 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. F
provides a proximity measure for graph vertices. By the matrix forest theorem,
F^{-1}=I+L, where L is the Laplacian matrix of G. We show that for the paths
and the so-called T-caterpillars, some diagonal entries of F (which provides a
measure of 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.
