3-partite Turán graphs due to edge deletion the change of distance signless Laplacian energy

Graph theory, which began with Euler's work, or more commonly known as the Königsberg Bridge Problem, has a wide range of applications. It is used in modeling across various fields such as physics, chemistry, computer science, economics, and engineering. Additionally, graphs are employed in solving problems like the shortest path problem, the four-color problem, the traveling salesman problem, and the facility location problem. In graphs, various matrices such as the adjacency matrix, distance matrix, distance signless Laplacian matrix, and generalized distance matrix are defined by considering the distances and degrees of vertices. The eigenvalues of these and similar matrices are used to define different types of energy, which are utilized to study various properties of structures modeled by graphs. Graphs have both practical applications and theoretical research topics. The most common theoretical research topic is the concept of energy. The energy concept in graphs was first defined by Ivan Gutman in 1978. Subsequently, various types of energy have been introduced in the literature. One of these, which is the main subject of this thesis, is the distance signless Laplacian energy, defined by Alhevaz and colleagues in 2018. Another widely studied topic in the literature is the energy change resulting from the deletion of an edge from a graph. This topic, initially studied by Day and So, has been explored for different types of energy and different types of graphs and continues to be an active area of research. This thesis consists of four main sections. The first section provides a brief overview of the origins of graph theory, its applications, the distance matrix, the distance signless Laplacian matrix, and edge deletion. This section also includes fundamental definitions that will aid our thesis work. The second section covers the literature review. The third section presents our main results, focusing first on the distance signless Laplacian energy change resulting from edge deletion in k-partite Kp,...,p complete graphs, followed by Kp,p,p+1 and Kp,p,p−1 complete graphs. The final section includes conclusions and recommendations.