Multiplicative Sombor index of trees

Abstract

For a graph $\Omega$, the multiplicative Sombor index is defined as $$\prod_{SO}(\Omega)=\prod_{ab\in \mathcal{E}(\Omega)}\sqrt{d^2_\Omega(a)+d^2_\Omega(b)},$$ where $d_\Omega(a)$ is the degree of vertex $a$. Liu [Liu, H. (2022). <em>Discrete Mathematics Letters</em>, 9, 80–85] showed that, when $\mathcal{T}$ is a tree of order $n$, $\prod_{SO}(\mathcal{T})\geqslant \prod_{SO}(P_n)=5(\sqrt{8})^{n-3}$. We improved this result and show that, if $\mathcal{T}$ is a tree of order $n$ with maximum degree $\cal{D}$, then $$\prod_{SO}(\mathcal{T})\geqslant \left\{\begin{array}{ll} (5({\cal{D}}^2+4))^{\frac{\cal{D}}{2}}8^{\frac{n-2{\cal{D}}-1}{2}} & {\rm if}\;{\cal{D}}\leqslant\frac{n-1}{2},\\[2mm] ({\cal{D}}^2+1)^{\frac{2{\cal{D}}+1-n}{2}}(5({\cal{D}}^2+4))^{\frac{n-{\cal{D}}-1}{2}} & {\rm if}\;{\cal{D}}>\frac{n-1}{2}. \end{array}\right. $$ Also, we show that equality holds if and only if $\mathcal{T}$ is a spider whose all legs have length less than three or all legs have length more than one.