  The eigenvalue problem of a graph Laplacian matrix $L$ arising from a simple,
connected and undirected graph has been given more attention due to its
extensive applications, such as spectral clustering, community detection,
complex network, image processing and so on. The associated graph Laplacian
matrix is symmetric, positive semi-definite, and is usually large and sparse.
Computing some smallest positive eigenvalues and corresponding eigenvectors is
often of interest.
  However, the singularity of $L$ makes the classical eigensolvers inefficient
since we need to factorize $L$ for the purpose of solving large and sparse
linear systems exactly. The next difficulty is that it is usually time
consuming or even unavailable to factorize a large and sparse matrix arising
from real network problems from big data such as social media transactional
databases, and sensor systems because there is in general not only local
connections.
  In this paper, we propose an eignsolver based on the inexact residual Arnoldi
method together with an implicit remedy of the singularity and an effective
deflation for convergent eigenvalues. Numerical experiments reveal that the
integrated eigensolver outperforms the classical Arnoldi/Lanczos method for
computing some smallest positive eigeninformation provided the LU factorization
is not available.
