Further results on almost controllable graphs

An eigenvalue lambda of a graph Gof order n is a main eigenvalue if its eigenspace is not orthogonal to the all-ones vector j. In 1978, Cvetkovic proved that Ghas exactly one main eigenvalue if and only if Gis regular, and posed the following long-standing problem: characterize the graphs with exactly k(2 <= k <= n) main eigenvalues. Graphs of order n with n, n - 1main eigenvalues are called controllable, almost controllable, respectively. In this paper, we study the properties of almost controllable graphs. For almost controllable trees, unicyclic and bicyclic graphs, we show that the diameters of their complements are less than or equal to 3, determine all complements with diameter 3, and obtain the results about the controllable graphs. Moreover, all integral almost controllable graphs are determined, and some further problems are proposed. (c) 2023 Elsevier Inc. All rights reserved.