On graphs with three eigenvalues

Graphs with four distinct Laplacian eigenvalues

“…It is well known that the class of all regular graphs with three distinct adjacency (Laplacian) eigenvalues coincides with the class of strongly regular graphs. For results on nonregular graphs with three adjacency eigenvalues, we refer the reader to [1,3,5,16]. Regular graphs with four adjacency (Laplacian) eigenvalues were studied in [6,10], and nonregular bipartite graphs with four adjacency eigenvalues were investigated in [8,9], through the study of the incidence graphs of some combinatorial designs.…”

A note on graphs whose signless Laplacian has three distinct eigenvalues

“…Regular graphs with three distinct A-eigenvalues (L-eigenvalues, Q-eigenvalues) are precisely strongly regular graphs and therefore graphs with three distinct eigenvalues can be considered as a generalization of strongly regular graphs. For results on graphs with few distinct A-eigenvalues, we refer the reader to [1,2,3,6,7,8,10,12] and on graphs with few distinct L-eigenvalues to [9,13]. In this paper, we investigate graphs with three distinct Q-eigenvalues and show that the largest Q-eigenvalue of a connected graph G is noninteger if and only if G = K n − e for n ≥ 4.…”

Graphs with two main and two plain eigenvalues

“…A connected regular graph with at most three distinct eigenvalues is known to be strongly regular, and see, for example [3] for a survey on strongly regular graphs. The connected non-regular graphs with three distinct eigenvalues have been studied by, for example, De Caen, Van Dam & Spence [7], Bridges & Mena [1], Muzychuk & Klin [13], Van Dam [17] and Cheng, Gavrilyuk, Greaves & Koolen [6]. The connected regular graphs with four distinct eigenvalues were studied by Van Dam [18], Van Dam & Spence [19] and Huang & Huang [12], among others.…”