We show that if μj is the j-th largest Laplacian eigenvalue, and dj is the j-th largest degree (1 j n) of a connected graph 蚠 on n vertices, then μj dj − j + 2 (1 j n − 1). This settles a conjecture due to Guo.

We give a sufficient condition for a distance-regular graph to be Hamiltonian. In particular, the Petersen graph is the only connected non-Hamiltonian strongly regular graph on fewer than 99 vertices.

AMS classification: 05C50, 05C70, 05E30.

We give some necessary conditions for a graph to be 3-chromatic in terms of the spectrum of the adjacency matrix.For all known distance-regular graphs it is determined whether they are 3-chromatic.A start is made with the classification of 3-chromatic distance-regular graphs, and it is shown...

2010 Mathematics Subject Classification: 05E30, 05C50;