Lebid V. Spectral analysis of infinite graphs

The thesis is devoted to study the spectral properties of infinite graphs with one and many infinite chains. The complete spectral analysis of infinite graphs with a single infinite chain was conducted firstly in the thesis. The operator A defined in l_2 (N) is associated to the graph. Spectral analysis graph is reduced to spectral analysis of the operator A. Investigated the discrete spectrum of the operator A, obtained criterion of existing of finite eigenvectors. Determined that the absolutely continuous spectrum is simple, and fills the interval [-2,2], and the singular continuous spectrum is absent. An explicit form of the spectral measure was received. Analysis of countable graphs with many rays contained in the thesis, carried out for certain classes of graphs: star, complete and cyclical. We prove that the operator A, which corresponds to this graph, unitarily equivalent to the orthogonal sum of finite symmetric matrices and Jacobi matrices.