On Topological Indices And Domination Numbers Of Graphs

Topological indices and dominating problems are popular topics in Graph Theory. There are various topological indices such as degree-based topological indices, distance-based topological indices and counting related topological indices et al. These topological indices correlate certain physicochemical properties such as boiling point, stability of chemical compounds. The concepts of domination number and independent domination number, introduced from the mid-1860s, are very fundamental in Graph Theory. In this dissertation, we provide new theoretical results on these two topics. We study k-trees and cactus graphs with the sharp upper and lower bounds of the degree-based topological indices(Multiplicative Zagreb indices). The extremal cacti with a distance-based topological index (PI index) are explored. Furthermore, we provide the extremal graphs with these corresponding topological indices. We establish and verify a proposed conjecture for the relationship between the domination number and independent domination number. The corresponding counterexamples and the graphs achieving the extremal bounds are given as well

On the Aα-spectral radii of cactus graphs

© 2020 by the authors. Let A(G) be the adjacent matrix and D(G) the diagonal matrix of the degrees of a graph G, respectively. For 0 ≤ α ≤ 1, the Aα-matrix is the general adjacency and signless Laplacian spectral matrix having the form of Aα(G) = αD(G) + (1-α)A(G). Clearly, A0(G) is the adjacent matrix and 2A1/2 is the signless Laplacian matrix. A cactus is a connected graph such that any two of its cycles have at most one common vertex, that is an extension of the tree. The Aα-spectral radius of a cactus graph with n vertices and k cycles is explored. The outcomes obtained in this paper can imply some previous bounds from trees to cacti. In addition, the corresponding extremal graphs are determined. Furthermore, we proposed all eigenvalues of such extremal cacti. Our results extended and enriched previous known results