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Let G be a simple graph. An eigenvalue of G, is a main eigenvalue if the corresponding eigenspace has a non-orthogonal eigenvector to the all-one vector j. A signed graph is a graph with a sign to each edge. If in the adjacency matrix of background graph change elements that corresponded by -1, set -1 and in the other elements don’t make any change, then we reach the sign matrix of a signed graph. By an eigenvalue of a signed graph, we mean an eigenvalue of its sign matrix. In this research, we study main eigenvalues of graphs and signed graphs. At first, we present the necessary and sufficient conditions for any graph which has exactly m main eigenvalues. Then, by introducing and creating... 

Let G be a graph of order n. The signless Laplacian matrix or Q-matrix of G is Q(G)=D(G)+A(G), where A(G) is the adjacency matrix of G and D(G) is diagonal degree matrix of G. The signless Laplacian characteristic polynomial or Q-polinomial of G is QG(x)=det(xI-Q(G)) and its roots are called signless Laplacian eigenvalues or Q-eigenvalues of G. If R is vertex-degree incidence matrix of G, then Q(G)=RRt. So Q(G) is a positive semi-definite matrix, i.e. its eigenvalues are none-negative. Let q1(G)≥q2(G)≥…≥qn(G) denote the signless Laplacian eigenvalues of G. A theory in which graphs are studied by means of eigenvalues of Q(G) is called signless Laplaciian theory or Q-theory.In this research,... 

Graph signals are sets of values residing on sets of nodes that are connected via edges. Graph Neural Networks (GNNs) are a type of machine learning model for working with graph-structured data, such as graph signals. GNNs have applications in graph classification, node classification, and link prediction. They can be thought of as learnable filters. In this thesis, our focus is on graph filters and enhancing the performance of GNNs. In the first part, we aim to reduce computational costs in graph signal processing, particularly in graph filters. We explore methods to transform signals to the frequency domain with lower computational cost. In the latter part, we examine regulations in...