  Nowadays a great attention has been focused on the discrete fractional
Laplace operator as the natural counterpart of the continuous one. In this
paper, we discretize the fractional Laplace operator $(-\Delta)^{s}$ for an
arbitrary finite graph and any positive real number $s$. It is shown that
$(-\Delta)^{s}$ can be explicitly represented by eigenvalues and eigenfunctions
of the Laplace operator $-\Delta$. Moreover, we study its important properties,
such as $(-\Delta)^{s}$ converges to $-\Delta$ as $s$ tends to $1$; while
$(-\Delta)^{s}$ converges to the identity map as $s$ tends to $0$ on a specific
function space. For related problems involving the fractional Laplace operator,
we consider the fractional Kazdan-Warner equation and obtain several existence
results via variational principles and the method of upper and lower solutions.
