On Unicyclic Reflexive Graphs

If G is a simple graph (a non-oriented graph without loops or multiple edges), its (0, 1)-adjacency matrix A is symmetric and roots of the characteristic polynomial PG (λ) = det (λI −A) (the eigenvalues of G, making up its spectrum) are all real numbers, for which we assume their non-increasing order: λ1 ≥ λ2 ≥ · · · ≥ λn. In a connected graph for the largest eigenvalue λ1 (the index of the graph) λ1 > λ2 holds, which need not take place otherwise, since the spectrum of a disconnected graph is the union of spectra of its components. The interrelation between the spectra of a graph and its induced subgraphs is established by the interlacing theorem: Let λ1 ≥ λ2 ≥ · · · ≥ λn be the eigenvalues of a graph G and μ1 ≥ μ2 ≥ ... ≥ μm eigenvalues of its induced subgraph H. Then the inequalities λn−m+i ≤ μi ≤ λi (i = 1, . . . ,m) hold. Thus e.g. if m = n − 1, λ1 ≥ μ1 ≥ λ2 ≥ μ2, . . ., and also λ1 > μ1 if G is connected. Reflexive graphs are graphs having λ2 ≤ 2. They correspond to some sets of vectors in the Lorentz space R and have some applications to the construction and classification of reflection groups [7]. Reflexive graphs that have been investigated so far are trees [4], [6], some classes of bicyclic graphs [10], [13] (see also [8]) and various classes of cactuses with more than two cycles [5], [9], [11], [12].