  The smallest eigenvalues and the associated eigenvectors (i.e., eigenpairs)
of a graph Laplacian matrix have been widely used in spectral clustering and
community detection. However, in real-life applications the number of clusters
or communities (say, $K$) is generally unknown a-priori. Consequently, the
majority of the existing methods either choose $K$ heuristically or they repeat
the clustering method with different choices of $K$ and accept the best
clustering result. The first option, more often, yields suboptimal result,
while the second option is computationally expensive. In this work, we propose
an incremental method for constructing the eigenspectrum of the graph Laplacian
matrix. This method leverages the eigenstructure of graph Laplacian matrix to
obtain the $K$-th smallest eigenpair of the Laplacian matrix given a collection
of all previously computed $K-1$ smallest eigenpairs. Our proposed method
adapts the Laplacian matrix such that the batch eigenvalue decomposition
problem transforms into an efficient sequential leading eigenpair computation
problem. As a practical application, we consider user-guided spectral
clustering. Specifically, we demonstrate that users can utilize the proposed
incremental method for effective eigenpair computation and for determining the
desired number of clusters based on multiple clustering metrics.
