For a (simple) graph G, by the neighborhood equivalence relation on G we mean the equivalence relation on V(G) given by : if and only if , where denotes the set of neighbors of u in G. An aim of this thesis is to support the view that the neighborhood equivalence relation is an important concept, which provides a vantage point to formulate, analyze or better understand results and problems in the spectral graph theory. Based on an old theorem of Haynsworth , which can also be ascribed to Goddard and Schneider, we first derive a new result on the spectrum of a block- stochastic matrix with the property that each off-diagonal block has equal entries and each diagonal block has equal diagonal entries and equal off-diagonal entries. Then we apply the result and the concept of reduced graph matrices to study the spectra of the usual graph matrices (including, in particular, the adjacency matrix, the Laplacian and the signless Laplacian) by partitioning the vertex set of the graph according to the neighborhood equivalence relation and make progress in the following five items : 1. The spectra of various graph matrices are given in terms of the spectra of the corresponding reduced graph matrices and certain multi-sets which depend on the cardinalities of the neighborhood equivalence classes and the common degree for vertices belonging to the same neighborhood equivalence class. 2. A formula for the characteristic polynomial of the signless Laplacian of a (degree) maximal graph is obtained and used to derive a (weak) interlacing relation between its reduced signless Laplacian eigenvalues and the degrees of its vertex. The signlsee Laplacian spectral radii of the well-known maximal graphs and (for ) are compared. 3. It is proved that a connected graph G of order n has n-2 as the second largest signless Laplacian eigenvalue if one of the following is satisfied : (a) G has at least two dominating vertices (which is the case if );(b) G has a pair of duplicate vertices with common degree n-2; (c) G is the graph obtained from by adding a pendant edge. It is also shown that for a connected graph of order , different from , if n-2 is a signless Laplacian eigenvalue, then the multiplicity of n-2 is at most n-2 with equality if and only if G=C(2,n-2) or n=4 and G=C4. 4. A set of necessary conditions for a maximal graph to have maximal signless Laplacian spectral radius among all connected graphs with given numbers of vertices and edges is obtained. 5. A new proof for the (known) description of the Laplacian spectrum of a maximal graph is offered.