@article{Çay2026, 
author = {Hatice Çay},
title = {Sombor index, polynomial, and energy of    m-graphs of finite abelian groups},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {7},
pages = {19749-19771},
keywords = {m-graph, Sombor index, forgotten index, Sombor energy, Sombor polynomial, finite abelian group, Euler Sombor index, average Sombor index},
url = {https://www.sciopen.com/article/10.3934/math.2026801},
doi = {10.3934/math.2026801},
abstract = {Let    H be a finite abelian group of order    n  ≥  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  ≅            Z        n   and the non-cyclic case    H  ≅            Z                      n        1              ⊕  ⋯  ⊕            Z                      n        r              ,  r  ≥  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        G    =      ∏          j      =      1              r        gcd  (      d          j        ,      q          j        ) governs the edge-type counts        N          B      S        =  D  −      G   and        N          A      S        =      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.}
}