The extremal unicyclic graphs with given diameter and minimum edge revised Szeged index

Let $H$ be a connected graph. The edge revised Szeged index of $H$ is defined as $Sz^{\ast}_{e}(H) = \sum\limits_{e = uv\in E_H}(m_{u}(e|H)+\frac{m_{0}(e|H)}{2})(m_{v}(e|H)+\frac{m_{0}(e|H)}{2})$, where $m_{u}(e|H)$ (resp., $m_{v}(e|H)$) is the number of edges whose distance to vertex $u$ (resp., $v$) is smaller than to vertex $v$ (resp., $u$), and $m_{0}(e|H)$ is the number of edges equidistant from $u$ and $v$. In this paper, the extremal unicyclic graphs with given diameter and minimum edge revised Szeged index are characterized

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