  The main result of this paper states that in a rooted product of a path with
rooted graphs which are disposed in a somewhat mirror-symmetric fashion, there
are distinct eigenvalues supported in the end vertices of the path which are
too close to each other: their difference is smaller than the square root of
two in the even distance case, and smaller than one in the odd distance case.
As a first application, we show that these end vertices cannot be involved in a
quantum walk phenomenon known as perfect state transfer, significantly
strengthening a recent result by two of the authors along with Godsil and van
Bommel. For a second application, we show that there is no balanced integral
tree of odd diameter bigger than three, answering a question raised by H\'{i}c
and Nedela in 1998. Our main technique involves manipulating ratios of
characteristic polynomials of graphs and subgraphs into continued fractions,
and exploring in detail their analytic properties. We will also make use of a
result due to P\'{o}lya and Szeg\"{o} about functions that preserve the
Lebesgue measure, which as far as we know is a novel application to
combinatorics. In the end, we connect our machinery to a recently introduced
algorithm to locate eigenvalues of trees, and with our approach we show that
any graph which contains two vertices separated by a unique path that is the
subdivision of a bridge with at least six inner vertices cannot be integral. As
a minor corollary this implies that most trees are not integral, but we believe
no one thought otherwise.
