AMS classifications; 05C50; 05E30;

AMS classifications: 05E30; 05B20

In this paper, we begin the determination of all primitive strongly regular graphs with chromatic number equal to 5.Using eigenvalue techniques, we show that there are at most 43 possible parameter sets for such a graph.For each parameter set, we must decide which strongly regular graphs, if...

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 classifications: 05C50; 05E99;

There are eleven integral trees with largest eigenvalue 3.

AMS Mathematics Subject Classification: 05C50.

The present article is designed to be a contribution to the chapter `Combinatorial Matrix Theory and Graphs' of the Handbook of Linear Algebra, to be published by CRC Press. The format of the handbook is to give just definitions, theorems, and examples; no proofs. In the five sections given...