Introduction/purpose: The Euler-Sombor index (EU) is a new vertexdegree-based graph invariant, obtained by geometric consideration. It is closely related to the Sombor index (SO). The actual form of ...this relation is established. Methods: Combinatorial graph theory is applied. Results: The inequalities between EU and SO are established. Conclusion: The paper contributes to the theory of Sombor-index-like graph invariants.
Introduction/purpose: The entire topological indices (T Ient) are a class of graph invariants depending on the degrees of vertices and edges. Some general properties of these invariants are ...established. Methods: Combinatorial graph theory is applied. Results: A new general expression for T Ient is obtained. For triangle-free and quadrangle-free graphs, this expression can be significantly simplified. Conclusion: The paper contributes to the theory of vertex and edge degree-based graph invariants.
Introduction/purpose: The temperature of a vertex of a graph of the order n is defined as d/(n-d), where d is the vertex degree. The temperature variant of the Sombor index is investigated and ...several of its properties established. Methods: Combinatorial graph theory is applied. Results: Extremal values and bounds for the temperature Sombor index are obtained. Conclusion: The paper contributes to the theory of Sombor-index-like graph invariants.
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.
Beyond the Zagreb indices Gutman, Ivan; Milovanović, Emina; Milovanović, Igor
AKCE International Journal of Graphs and Combinatorics,
01/2020, Volume:
17, Issue:
1
Journal Article
Peer reviewed
Open access
The two Zagreb indices M1=∑vd(v)2 and M2=∑uvd(u)d(v) are vertex-degree-based graph invariants that have been introduced in the 1970s and extensively studied ever since. In the last few years, a ...variety of modifications of M1 and M2 were put forward. The present survey of these modified Zagreb indices outlines their main mathematical properties, and provides an exhaustive bibliography.
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.
The degree of a vertex of a molecular graph is the number of first neighbors of this vertex. A large number of molecular-graph-based structure descriptors (topological indices) have been conceived, ...depending on vertex degrees. We summarize their main properties, and provide a critical comparative study thereof. (doi: 10.5562/cca2294) Keywords: topological index, molecular structure descriptor, vertex-degree-based topological index, molecular graph, chemical graph theory
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