Further results on almost controllable graphs

Abstract

An eigenvalue λ of a graph G of order n is a main eigenvalue if its eigenspace is not orthogonal to the all-ones vector j. In 1978, Cvetković proved that G has exactly one main eigenvalue if and only if G is 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−1 main 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.