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articleFree Access
The $k$ $k$ -Sombor Index of Trees
- Fangxia Wang and
- Baoyindureng Wu
Asia-Pacific Journal of Operational Research21 Apr 2023
Preview Abstract
For a positive real number $k$ $k$ , the $k$ $k$ -Sombor index of a graph $G$ $G$ , introduced by Réti et al, is defined as
${SO}_{k} (G) = \sum_{u v \in E (G)} \sqrt[k]{d {(u)}^{k} + d {(v)}^{k}},$ ${SO}_{k} (G) = \sum_{u v \in E (G)} \sqrt[k]{d {(u)}^{k} + d {(v)}^{k}},$
where $d (u)$ $d (u)$ denotes the degree of the vertex $u$ $u$ in $G$ $G$ . By the definition, $S O_{2} (G)$ $S O_{2} (G)$ is exactly the Sombor index of $G$ $G$ , while $S O_{1} (G)$ $S O_{1} (G)$ is the first Zagreb index of $G$ $G$ .
In this paper, for $k \geq 1$ $k \geq 1$ we present the extremal values of the $k$ $k$ -Sombor index of trees with some given parameters, such as matching number, the number of pendant vertices, diameter. This generalizes the relevant results on Sombor index due to Chen, Li and Wang ((2002). Extremal values on the Sombor index of trees, MATCH Communications in Mathematical and in Computer Chemistry, 87, 23–49).
Handling $S O_{k} (G)$ $S O_{k} (G)$ appears to be different for $k < 1$ $k < 1$ in contrast to the case when $k \geq 1$ $k \geq 1$ . To demonstrate this, we also characterize the extremal trees with respect to the ${SO}_{\frac{1}{2}}$ ${SO}_{\frac{1}{2}}$ with matching number, the number of pendant vertices and diameter. In addition, three relevant conjectures are proposed.

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