Regular graphs with four eigenvalues

Abstract

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Graphs with four distinct Laplacian eigenvalues

“…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.…”

“…With an easy calculation, we obtain that D 1 and D 2 have the Laplacian spectra 0 [1] , 2 [6] , 5 [7] , 9 [1] and 0 [1] , 3 [6] , 6 [7] , 10 [1] , respectively.…”

Some constructions of integral graphs

“…We mention that Φ 0 (G) has been studied in [6] to characterize a family of graphs with specified eigenvalues. Combining Corollary 1 and Lemma 1, we obtain the following result.…”

Graphs with two main and two plain eigenvalues

“…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. Cioabă, Haemers, Vermette & Wong [4] (resp.…”