Laplacian Degrees of a Graph
(Reproduced from [M].)
Let be the Laplacian matrix of a graph of order . Let be the -th largest eigenvalue of (). For the purpose of this problem, we call the number
the -th Laplacian degree of . In addition to that, let be the -th largest (usual) degree in . It is known that every connected graph satisfies for [GM], [LP] and for [G].
[GM] R. Grone, R. Merris, The Laplacian spectrum of a graph II, SIAM J. Discrete Math.7 (1994) 221-229. MathSciNet
[LP] J.S. Li, Y.L. Pan, A note on the second largest eigenvalue of the Laplacian matrix of a graph, Linear Multilin. Algebra 48 (2000) 117-121. MathSciNet
*[G] J.-M. Guo, On the third largest Laplacian eigenvalue of a graph, Linear Multilin. Algebra 55 (2007) 93-102. MathSciNet
[M] B. Mohar, Problem of the Month
* indicates original appearance(s) of problem.