In this paper, we investigate the Sombor index of the total graph and unit graph of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {Z}_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> which is denoted by <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$T_{\Gamma }(\mathbb {Z}_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>T</mi> <mi mathvariant="normal">Γ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">Z</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$G(\mathbb {Z}_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">Z</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> respectively for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$n \in \{2k, p^{\alpha }, pq, p^2q\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mo stretchy="false">{</mo> <mn>2</mn> <mi>k</mi> <mo>,</mo> <msup> <mi>p</mi> <mi>α</mi> </msup> <mo>,</mo> <mi>p</mi> <mi>q</mi> <mo>,</mo> <msup> <mi>p</mi> <mn>2</mn> </msup> <mi>q</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> where p and q are distinct odd prime numbers such that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p < q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo><</mo> <mi>q</mi> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, we compute the Sombor index of any finite local ring. These results contribute to the study of degree-based topological indices in algebraic graph theory, and they also have applications in mathematical chemistry.

On the Sombor index of the total graph and the unit graph of commutative rings

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发表日期：

2026年6月30日

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Abhishek Vaibhav Pathak,
Anukul Sachan,
Raisa D’Souza

摘要

In this paper, we investigate the Sombor index of the total graph and unit graph of $\mathbb {Z}_n$ $Z_{n}$ which is denoted by $T_{\Gamma }(\mathbb {Z}_n)$ $T_{Γ} (Z_{n})$ and $G(\mathbb {Z}_n)$ $G (Z_{n})$ respectively for $n \in \{2k, p^{\alpha }, pq, p^2q\}$ $n \in {2 k, p^{α}, p q, p^{2} q}$ where p and q are distinct odd prime numbers such that $p < q$ $p < q$ . Furthermore, we compute the Sombor index of any finite local ring. These results contribute to the study of degree-based topological indices in algebraic graph theory, and they also have applications in mathematical chemistry.