Some applications of Laplace eigenvalues of graphs

Distributed average consensus with least-mean-square deviation

“…According to the numerical results in Section 5, we conjecture that the optimal solution is = 1/(d + 1), but we have not been able to prove this yet. The value = 1/(d + 1) is also the solution for the FDLA problem studied in [33] (see also [9,19,24]). …”

“…The formula (19) gives us the optimality conditions for the problem (18): a weight vector w is optimal if and only if F (w ) 0, G(w ) 0, and, for all {i, j } ∈ E, we have…”

“…. , m, we compute *f/*w k , using the last line of (19). For each k, this involves subtracting two columns of F −1 , and finding the norm squared of the difference, and the same for G −1 , which has a cost O(n), so this step has a total cost O(mn).…”

“…This mesh graph is the Cartesian products of two n-node rings (see, e.g., [19]). The Laplacian is the Kroneker product of two circulant matrices, and has eigenvalues…”

Graph matching and clustering using spectral partitions

“…the eigenvector associated with the second smallest eigenvalue of the Laplacian matrix. In a useful review, Mohar [10] has summarized some important applications of Laplace eigenvalues such as the max-cut problem, semidefinite programming and steady state random walks on Markov chains. More recently, Haemers [11] has explored the use of interlacing properties for the eigenvalues and has shown how these relate to the chromatic number, the diameter and the bandwidth of graphs.…”