Cacti with Extremal PI Index

The vertex PI index $PI(G) = \sum_{xy \in E(G)} [n_{xy}(x) + n_{xy}(y)]$ is a distance-based molecular structure descriptor, where $n_{xy}(x)$ denotes the number of vertices which are closer to the vertex $x$ than to the vertex $y$ and which has been the considerable research in computational chemistry dating back to Harold Wiener in 1947. A connected graph is a cactus if any two of its cycles have at most one common vertex. In this paper, we completely determine the extremal graphs with the largest and smallest vertex PI indices among all the cacti. As a consequence, we obtain the sharp bounds with corresponding extremal cacti and extend a known result.Comment: Accepted by Transactions on Combinatorics, 201

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