On the change of seidel energy of complete r−partite graphs due to edge deletion and normalized Seidel laplacian

Graphs are structures consisting of vertices and edges. Edges are modelled to connect the vertices of graphs. The best example of graph modelling is the representation of atoms in a molecule as vertices and bonds between atoms as edges. Thus, the properties of molecules are analysed using graphs. Depending on the vertex and edge relationship in graphs, many matrices such as adjacency matrix, Laplacian matrix, distance matrix, Seidel matrix have been defined. The eigenvalues of these matrices are used to find the graph energy, which is still being studied in the literature. The energy of a graph was first defined by Gutman in 1978. The sum of absolute values of the eigenvalues of the adjacency matrix of a graph G gives the energy of the graph. It is also known as molecular graph energy. It was developed to model the π−electron energy of molecules. As many matrices are defined in graphs, energy types are also defined. The type of energy studied in the thesis is the Seidel energy, which was first defined by Haemers in 2012. One of topics that has been studied intensively in the literature is the energy change of the graph when an edge (or vertex) is deleted from the graph. The main subject of our thesis is also shaped around this subject for Seidel energy. In the first chapter, general information about graph theory and its applications are given. In addition, in this chapter, basic definitions are also included. In the second chapter, a literature search is conducted and literature information is given. In the third chapter, the Seidel energy change for r−partite complete graphs Kp,··· ,p, Kp,p,··· ,p,p+1 and Kp,p,··· ,p,p−1, when an edge is deleted is analysed. In the fourth chapter, we define the normalized Seidel Laplacian matrix. Then the eigenvalue analysis of this matrix is carried out. Like other energy types, the normalized Seidel Laplacian energy is defined and lower and upper bounds are obtained. The last chapter, chapter five, contains the conclusions and recommendations of this thesis.