Optimal inequalities and extremal problems on the general Sombor index

Abstract

In this work we obtain new lower and upper optimal bounds of general Sombor indices. Specifically, we have inequalities for these indices relating them with other indices: the first Zagreb index, the forgotten index and the first variable Zagreb index. Finally, we solve some extremal problems for general Sombor indices.

Extremal polygonal cacti for bond incident degree indices

“…Finally, it is easy to see that (11) holds when α > 1 and β ≥ 1 by the monotonicity of s β + t β α . Therefore, s β + t β α is special escalating.…”

“…where α, β are real numbers. We note that the above form is a natural generalization of the Sombor index, which was also introduced elsewhere, e.g., the first (β, α) − KA index in [12] and the general Sombor index in [11]. In addition to the first Zagreb, Sombor and the α-Sombor index listed above, the general Sombor index also includes many indices, e.g., the modified first Zagreb index (α = 1, β = −3) [17], forgotten index (α = 1, β = 2) [6], inverse degree index (α = 1, β = −2) [5], modified Sombor index (α = −1/2, β = 2) [13], first Banhatti-Sombor index (α = 1/2, β = −2) [15] and general sum-connectivity index (α ∈ R, β = 1) [24].…”