OPTIMAL LINEAR LABELINGS AND EIGENVALUES OF GRAPHS

For several NP-hard optimal linear labeling problems, including the bandwidth, the cutwidth, and the min-sum problem for graphs, a heuristic algorithm is proposed which finds approximative solutions to these problems in polynomial time. The algorithm uses eigenvectors corresponding to the second smallest Laplace eigenvalue of a graph. Although bad in some "degenerate" cases, the algorithm shows fairly good behaviour. Several upper and lower bounds on the bandwidth, cutwidth, and min-p-sums are derived. Most of these bounds are given in terms of Laplace eigenvalues of the graphs. They are used in the analysis of our algorithm and as measures for the error of the obtained approximation to an optimal labeling.