A Survey on Integral Graphs

Throughout this paper a graph G is assumed to be simple, i.e. a finite undirected graph without loops or multiple edges. Therefore, the characteristic polynomial of (the adjacency matrix of) G, denoted by PG(λ), has only real zeroes and this family of eigenvalues (the spectrum of G) will be represented as (λ1, λ2, . . . , λn), where λ1 ≥ λ2 ≥ · · · ≥ λn or in the form μ1 1 , μ2 2 , . . . , μm m , where μ1, μ2, . . . , μm are distinct eigenvalues of G in decreasing order and k1, k2, . . . , km are the corresponding multiplicities. The sum ∑n i=1 λ k i is called the k-th spectral moment and is equal to the number of closed walks of length k of G. The characteristic polynomial of a graph is monic (i.e. its leading coefficient is 1), and hence the rational eigenvalues are integers. A graph whose spectrum consists entirely of integers is called an integral graph. Since there is no general characterization (besides the definition) of these graphs, the problem of finding (or characterizing) integral graphs has to be treated in some special classes of graphs. This text gives a survey of former investigations and main results concerning this topic. The paper is based on a chapter on the same subject of the book [54]. For all notation and terminology see [20, 54].