Sombor index, polynomial, and energy of $m$-graphs of finite abelian groups

Let $H$ be a finite abelian group of order $n\ge 2$ and let $m>1$ be an integer. The $m$-graph $m$- $G(H)$ is the simple undirected graph on vertex set $H$ in which two distinct elements $a$ and $b$ are adjacent if and only if $a^m=b$ or $b^m=a$. Building on the structural characterization of connected $m$-graphs established by Badawi, we investigated degree-based topological invariants of connected $m$- $G(H)$ in both the cyclic case $H\cong {\mathbb{Z}}_n$ and the non-cyclic case $H\cong {\mathbb{Z}}_{n_1}\oplus \cdots \oplus {\mathbb{Z}}_{n_r},r\ge 2.$ Using an edge-type decomposition arising from the degree structure of connected $m$-graphs, we derived explicit formulas for the Sombor index, Sombor polynomial, reduced and modified Sombor indices, Euler Sombor index, average Sombor index, and forgotten topological index. In the non-cyclic setting, the parameter $\mathcal{G}=\prod _{j=1}^{r}gcd(d_j,q_j)$ governs the edge-type counts $N_{BS}=D-\mathcal{G}$ and $N_{AS}=\mathcal{G}-1$, which cannot be expressed purely in terms of $D$ and $Q$. We also determined closed-form expressions for the Sombor energy in the cases $Q=1$ (star graphs) and $Q=2$ (two-hub trees). All theoretical results were independently verified by exact symbolic computation using SageMath 10.3 and the SymPy library.