In this paper, for $k \geq 1$ we present the extremal values of the $k$ -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 $S O_{k} (G)$ appears to be different for $k < 1$ in contrast to the case when $k \geq 1$ . To demonstrate this, we also characterize the extremal trees with respect to the ${SO}_{\frac{1}{2}}$ with matching number, the number of pendant vertices and diameter. In addition, three relevant conjectures are proposed.

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References

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