Topological indices, particularly the Sombor index, have emerged as powerful tools for quantifying molecular structure–property relationships. This study investigates the predictive capacity of Sombor indices across two distinct systems: a tetrahedral diamond lattice (TDL) and twenty-eight polychlorobiphenyl (PCB) compounds. For the diamond lattice, we derived exact closed-form polynomial expressions for the Sombor index (SO) and its variants (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$SO_3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>O</mi> <mn>3</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>–<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$SO_6$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>O</mi> <mn>6</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>) as functions of the lattice dimension r, revealing their sensitivity to geometric regularity. For PCBs, Sombor indices were calculated and correlated with key physicochemical properties (melting point, relative retention time, log P, heat of formation, Henry’s constant). Remarkably strong linear relationships were observed, with correlation coefficients (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$R^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>R</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>) exceeding 0.95 for relative retention time (RTT). Remarkably strong linear relationships were observed for relative retention time (RTT), with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$SO, SO_3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mi>O</mi> <mo>,</mo> <mi>S</mi> <msub> <mi>O</mi> <mn>3</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$SO_4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>O</mi> <mn>4</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> achieving <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$R^2>0.93$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>></mo> <mn>0.93</mn> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq7"> <EquationSource Format="TEX">$p<0.001$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo><</mo> <mn>0.001</mn> </mrow> </math></EquationSource> </InlineEquation>) exceeding 93% variance explanation while <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$SO_6$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>O</mi> <mn>6</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> yielded <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$R^2=0.873$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0.873</mn> </mrow> </math></EquationSource> </InlineEquation>. In contrast, correlations with melting point, log P, and Henry’s constant were significant for <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$SO, SO_3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mi>O</mi> <mo>,</mo> <mi>S</mi> <msub> <mi>O</mi> <mn>3</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$SO_4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>O</mi> <mn>4</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$SO_6$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>O</mi> <mn>6</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq13"> <EquationSource Format="TEX">$R^2=0.875-1.000$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0.875</mn> <mo>-</mo> <mn>1.000</mn> </mrow> </math></EquationSource> </InlineEquation>) but weaker for <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$SO_5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>O</mi> <mn>5</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq15"> <EquationSource Format="TEX">$R^2 =0.319$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0.319</mn> </mrow> </math></EquationSource> </InlineEquation>). These results underscore the versatility of Sombor-type descriptors in bridging molecular topology with experimental behavior, offering a computationally efficient strategy for property prediction in materials design and environmental risk assessment.

On the predictive power of Sombor indices: from diamond lattices to polychlorinated biphenyls

期刊论文
发表日期：

2025年10月4日

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Song Tingting,
Sadia Noureen,
Amna Maryam,
Adnan Aslam

摘要

Topological indices, particularly the Sombor index, have emerged as powerful tools for quantifying molecular structure–property relationships. This study investigates the predictive capacity of Sombor indices across two distinct systems: a tetrahedral diamond lattice (TDL) and twenty-eight polychlorobiphenyl (PCB) compounds. For the diamond lattice, we derived exact closed-form polynomial expressions for the Sombor index (SO) and its variants ( $SO_3$ $S O_{3}$ – $SO_6$ $S O_{6}$ ) as functions of the lattice dimension r, revealing their sensitivity to geometric regularity. For PCBs, Sombor indices were calculated and correlated with key physicochemical properties (melting point, relative retention time, log P, heat of formation, Henry’s constant). Remarkably strong linear relationships were observed, with correlation coefficients ( $R^2$ $R^{2}$ ) exceeding 0.95 for relative retention time (RTT). Remarkably strong linear relationships were observed for relative retention time (RTT), with $SO, SO_3$ $S O, S O_{3}$ , and $SO_4$ $S O_{4}$ achieving $R^2>0.93$ $R^{2} > 0.93$ ( $p<0.001$ $p < 0.001$ ) exceeding 93% variance explanation while $SO_6$ $S O_{6}$ yielded $R^2=0.873$ $R^{2} = 0.873$ . In contrast, correlations with melting point, log P, and Henry’s constant were significant for $SO, SO_3$ $S O, S O_{3}$ , $SO_4$ $S O_{4}$ , and $SO_6$ $S O_{6}$ ( $R^2=0.875-1.000$ $R^{2} = 0.875 - 1.000$ ) but weaker for $SO_5$ $S O_{5}$ ( $R^2 =0.319$ $R^{2} = 0.319$ ). These results underscore the versatility of Sombor-type descriptors in bridging molecular topology with experimental behavior, offering a computationally efficient strategy for property prediction in materials design and environmental risk assessment.