Suchergebnisse

Introduction/purpose: Vertex-degree-based (VDB) graph matrices form a special class of matrices, corresponding to the currently much investigated vertex-degree-based (VDB) graph invariants. Some spectral properties of these matrices are investigated. Results: Generally valid sharp lower and upper bounds are established for the spectral radius of any VDB matrix. The equality cases are characterized. Several earlier published results are shown to be special cases of the presently reported bounds. Conclusion: The results of the paper contribute to the general spectral theory of VDB matrices, as well as to the general theory of VDB graph invariants.

Introduction/purpose: In the current literature, several dozens of vertex-degree-based (VDB) graph invariants are being studied. To each such invariant, a matrix can be associated. The VDB energy is the energy (= sum of the absolute values of the eigenvalues) of the respective VDB matrix. The paper examines some general properties of the VDB energy of bipartite graphs. Results: Estimates (lower and upper bounds) are established for the VDB energy of bipartite graphs in which there are no cycles of size divisible by 4, in terms of ordinary graph energy. Conclusion: The results of the paper contribute to the spectral theory of VDB matrices, especially to the general theory of VDB energy.

Introduction/purpose: The Sombor matrix is a vertex-degree-based matrix associated with the Sombor index. The paper is concerned with the spectral properties of the Sombor matrix. Results: Equalities and inequalities for the eigenvalues of the Sombor matrix are obtained, from which two fundamental bounds for the Sombor energy (= energy of the Sombor matrix) are established. These bounds depend on the Sombor index and on the "forgotten" topological index. Conclusion: The results of the paper contribute to the spectral theory of the Sombor matrix, as well as to the general spectral theory of matrices associated with vertex-degree-based graph invariants.

Introduction/purpose: The paper presents numerous vertex-degree-based graph invariants considered in the literature. A matrix can be associated to each of these invariants. By means of these matrices, the respective vertex-degree-based graph energies are defined as the sum of the absolute values of the eigenvalues. Results: The article determines the conditions under which the considered graph energies are greater or smaller than the ordinary graph energy (based on the adjacency matrix). Conclusion: The results of the paper contribute to the theory of graph energies as well as to the theory of vertex-degree-based graph invariants.

In addition to a great variety of degree-based and distance-based molecular structure descriptors, there are a few degree-and-distance-based topological indices. Two main such indices are the degree distance (DD) and the Gutman index (ZZ). Their mutual relations are analyzed and several new such relations established. It is shown that by conveniently chosen linear combinations of DD, ZZ and the Wiener index, it is possible to calculate several chemically interesting structural properties of molecular graphs.