We consider regular graphs with small second largest eigenvalue (denoted by λ2). In particular, we determine all triangle-free regular graphs with λ2 ≤ √ 2, all bipartite regular graphs with λ2 ≤ √ 3, and all bipartite regular graphs of degree 3 with λ2 ≤ 2.2010 Mathematics Subject Classification. 05C50.

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The Q-index of a simple graph is the largest eigenvalue of its signless Laplacian. As for the adjacency spectrum, we will show that in the set of connected bipartite graphs with fixed order and size, the bipartite graphs with maximal Q-index are the double nested graphs. We provide a sequence of (in)equalities regarding the principal eigenvector of the signless Laplacian of double nested graphs and apply these results to obtain some lower and upper bounds for their Q-index. In the end, we give some computational results in order to compare these bounds.

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