  In spectral bisection, a Fielder vector is used for partitioning a graph into
two connected subgraphs according to its sign pattern. In this article, we
investigate graphs having Fiedler vectors with unbalanced sign patterns such
that a partition can result in two connected subgraphs that are distinctly
different in size. We present a characterization of graphs having a Fiedler
vector with exactly one negative component, and discuss some classes of such
graphs. We also establish an analogous result for regular graphs with a Fiedler
vector with exactly two negative components. In particular, we examine the
circumstances under which any Fiedler vector has unbalanced sign pattern
according to the number of vertices with minimum degree.

Combinatorics;Spectral Theory