  This paper presents the applications of Eigenvalues and Eigenvectors (as part
of spectral decomposition) to analyze the bipartivity index of graphs as well
as to predict the set of vertices that will constitute the two partitions of
graphs that are truly bipartite and those that are close to being bipartite.
Though the largest eigenvalue and the corresponding eigenvector (called the
principal eigenvalue and principal eigenvector) are typically used in the
spectral analysis of network graphs, we show that the smallest eigenvalue and
the smallest eigenvector (called the bipartite eigenvalue and the bipartite
eigenvector) could be used to predict the bipartite partitions of network
graphs. For each of the predictions, we hypothesize an expected partition for
the input graph and compare that with the predicted partitions. We also analyze
the impact of the number of frustrated edges (edges connecting the vertices
within a partition) and their location across the two partitions on the
bipartivity index. We observe that for a given number of frustrated edges, if
the frustrated edges are located in the larger of the two partitions of the
bipartite graph (rather than the smaller of the two partitions or equally
distributed across the two partitions), the bipartivity index is likely to be
relatively larger.
