The estrada index in graphene structure

The Estrada Index, defined by the eigenvalues of the adjacency matrix of graphs, defines the structural centrality measure of graphs. In the first part of the thesis, an introduction to the usage areas of graph theory is given, and in the second part, the basic definitions about graph theory are given. The third part of the thesis includes a summary of references, and the fourth part covers materials and methods. The fifth part examines graphene structures in graph theory using the Estrada Index, and some lower and upper bounds for this structure are obtained. Graphene, which possesses surprising chemical and physical properties, is analyzed in terms of its bipartite graph and cycle graph properties, and the Estrada Index is calculated. By examining certain theorems and lemmas, new inequalities related to the Estrada Index are derived. In these inequalities, concepts such as the number of nodes in graphene sheets, the maximum and minimum degrees of nodes, the number of edges, and their zeros are evaluated. The aim of the study is to utilize the Estrada Index to benefit the mathematical modeling of structures such as graphene and to introduce innovations in methods applied in the field of molecular structure. Keywords: Graph, Estrada Index, Graphene.