Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$G=(V,E)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a graph with the vertex set V(G) and edge set E(G). The Sombor index of G, SO(G), is defined as <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\sum _{uv\in E(G)} \sqrt{deg(u)^{2}+deg(v)^{2}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>∑</mo> <mrow> <mi>u</mi> <mi>v</mi> <mo>∈</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </msub> <msqrt> <mrow> <mi>d</mi> <mi>e</mi> <mi>g</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <mi>d</mi> <mi>e</mi> <mi>g</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </math></EquationSource> </InlineEquation>, where deg(u) is the degree of vertex u in V(G). The intersection graph of ideals of a commutative ring R consists of the set of all non-trivial ideals as the vertex set and two distinct vertices I, J are joined by an edge if and only if <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$I\small \cap J \ne 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>I</mi> <mstyle mathsize="0.6em"> <mo>∩</mo> </mstyle> <mi>J</mi> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. In this article, we investigate the Sombor index and Sombor spectrum of the graph <InlineEquation ID="IEq4"> <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>, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n\in \mathbb {N}.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation>

Sombor index in intersection graphs of ideals of rings

期刊论文
发表日期：

2026年2月3日

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R. Nikandish,
M. Mehrara,
M. J. Nikmehr

摘要

Let $G=(V,E)$ $G = (V, E)$ be a graph with the vertex set V(G) and edge set E(G). The Sombor index of G, SO(G), is defined as $\sum _{uv\in E(G)} \sqrt{deg(u)^{2}+deg(v)^{2}}$ $\sum_{u v \in E (G)} \sqrt{d e g {(u)}^{2} + d e g {(v)}^{2}}$ , where deg(u) is the degree of vertex u in V(G). The intersection graph of ideals of a commutative ring R consists of the set of all non-trivial ideals as the vertex set and two distinct vertices I, J are joined by an edge if and only if $I\small \cap J \ne 0$ $I \cap J \neq 0$ . In this article, we investigate the Sombor index and Sombor spectrum of the graph $G(\mathbb {Z}_{n})$ $G (Z_{n})$ , where $n\in \mathbb {N}.$ $n \in N .$