Properties of generalized Frank matrices

In this study, we have defined generalized Frank matrix based on the Frank matrix and have computed the determinant, inverse, characteristic polynomial of this matrix. Using the Sturm Theorem, we obtained that all eigenvalues of generalized Frank matrices are different and positive under certain conditions. We examined the special cases of generalized Frank matrices and the results we found with the Sturm Theorem. We have defined the Frank graph concept which its adjacency matrix equals to the Frank matrix and additionally concept of the Frank energy. We obtained some bounds for spectral radius of the Frank graph and also Frank energy. This study consists of four sections. In the first section, the purpose of our study has been determined and a summary of the literature is included. In the second section, some basic concepts about our study, Frank matrix and its properties, and basic informations about Sturm series are included. In the third section, generalized Frank matrices and their algebraic properties, the application of the Sturm Theorem to the generalized Frank matrices, some applications involving the special cases of the generalized Frank matrix are given. In addition, the applications of the Frank matrix in graph theory, some bounds found for the spectral radius and energy of the Frank graph are also included in the third section. The fourth section consists of results and suggestions.