Ranks and Representations for Spectral Graph Bisection

Abstract

We investigate the minimum graph bisection size according to spectral decompositions of adjacency matrices in several representations (e.g., Laplacian, 01 adjacency, Seidel). In addition, we introduce a class of adjacency matrix representations by including a multiplicative vertex degree along the diagonal of the Seidel adjacency matrix. The new representation, denoted $S(c)$, often provided smaller spectral bisections in comparison with the other representations for certain classes of graphs. By adjusting the multiplicative parameter $c$, the cut size performance varied predictably between the Seidel, 01 adjacency, and Laplacian representations for both random graphs and random geometric graphs with various edge probability parameters. Furthermore, we analyzed the cut sizes over many representations according to each eigenvector rank for large classes of random and real world graphs to give experimental evidence that the quality of spectral bisections depends on the eigenvector rank, representation, and graph type. Ultimately, we provide evidence that particular ranks and representations are predictably more appropriate for use with certain types of graphs. In general, the related Laplacian and $S(2.0)$ representations produced the smallest cuts for geometric graphs, whereas the Seidel, 01 adjacency, and $S(c)$ (with $0\leq c\leq1$) produced the smallest cuts for random graphs. We anticipate that our results will provide motivation for a unifying theorem of spectral bisection across eigenvector ranks, adjacency representations, and graph types.