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In spectral bisection, a Fielder vector is used for partitioning a graph into two connected subgraphs according to its sign pattern. 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...
We investigate the properties of the least-squares solution of the system of equations with a matrix being the incidence matrix of a given undirected connected graph and we propose an algorithm that uses this solution for finding a vertex-disjoint cycle cover (2-factor) of the graph .
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