  As a measure of vertex importance according to the graph structure, PageRank
has been widely applied in various fields. While many PageRank algorithms have
been proposed in the past decades, few of them take into account whether the
graph under investigation is directed or not. Thus, some important properties
of undirected graph\textemdash symmetry on edges, for example\textemdash is
ignored. In this paper, we propose a parallel PageRank algorithm specifically
designed for undirected graphs that can fully leverage their symmetry.
Formally, our algorithm extends the Chebyshev Polynomial approximation from the
field of real function to the field of matrix function. Essentially, it
reflects the symmetry on edges of undirected graph and the density of
diagonalizable matrix. Theoretical analysis indicates that our algorithm has a
higher convergence rate and requires less computation than the Power method,
with the convergence rate being up to 50\% higher with a damping factor of
$c=0.85$. Experiments on six datasets illustrate that our algorithm with 38
parallelism can be up to 39 times faster than the Power method.
