




(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].
Bibliography
[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.