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articleNo Access
Maximum and minimum Sombor index among $k$ -apex unicyclic graphs and $k$ -apex trees
- Jing Yang and
- Hanyuan Deng
Asian-European Journal of Mathematics11 Jun 2022
Preview Abstract
The Sombor index $SO$ of a graph $G$ is defined as
$SO (G) = \sum_{u v \in E (G)} \sqrt{{d_{G} (u)}^{2} + {d_{G} (v)}^{2}},$
where $d_{G} (v)$ is the degree of the vertex $v$ of $G$ . A $k$ -cone $c$ -cyclic graph is the join of the complete graph $K_{k}$ and a connected $c$ -cyclic graph. A $k$ -apex tree (respectively, $k$ -apex unicyclic graph) is defined as a connected graph $G$ with a $k$ -subset $V_{k} \subseteq V (G)$ such that $G - V_{k}$ is a tree (respectively, unicyclic graph), but $G - X$ is not a tree (respectively, unicyclic graph) for any $X \subseteq V (G)$ with $| X | < k$ . In this paper, we show the minimal graphs of $SO (G)$ among all $k$ -cone $c$ -cyclic graphs with $π$ as their degree sequence, and determine the extremal values and extremal graphs of $SO (G)$ among $k$ -apex unicyclic graphs and $k$ -apex trees, respectively.
articleNo Access
On general Sombor index of graphs
Asian-European Journal of Mathematics07 Sep 2022
Preview Abstract
In this paper, we extend the recently introduced vertex-degree-based topological index, the Sombor index, and we call it general Sombor index. The general Sombor index generalizes both the forgotten index and the Sombor index. We present the bounds in terms of other important graph parameters for general Sombor index. We also explore the Nordhaus–Gaddum-type result for the general Sombor index. We present further the relations between general Sombor index and other generalized indices: general Randić index and general sum-connectivity index.