Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 XuIndex

Source Notebook

Compute the Xu index of an undirected graph or a molecule

Contributed by: Jan Mangaldan

ResourceFunction["XuIndex"][g]

computes the Xu index of the graph g.

ResourceFunction["XuIndex"][mol]

computes the Xu index of the molecule mol.

Details

For an undirected connected graph with n vertices, the Xu index is defined as , where δi is the degree of the ith vertex, and σi is the ith row sum of the graph distance matrix.
ResourceFunction["XuIndex"][mol] computes the Xu index of a molecule mol, where the hydrogens are ignored by default. Use the option setting IncludeHydrogens→All to account for hydrogens.
ResourceFunction["XuIndex"][entity] computes the Xu index of an entity of type "Chemical" or "Graph".

Examples

Basic Examples (1) 

The Xu index of a Petersen graph:

In[1]:=
ResourceFunction["XuIndex"][PetersenGraph[]]
Out[1]=

Scope (2) 

Compute the Xu index of a molecule:

In[2]:=
ResourceFunction["XuIndex"][ Molecule[{"C", "C", "C", "C", "C", "C", "C", "H", "H", "H", "H", "H", "H", "H", "H", "H", "H", "H", "H"}, { Bond[{1, 2}, "Single"], Bond[{2, 3}, "Single"], Bond[{3, 4}, "Single"], Bond[{4, 5}, "Single"], Bond[{5, 6}, "Single"], Bond[{6, 7}, "Single"], Bond[{6, 1}, "Single"], Bond[{7, 3}, "Single"], Bond[{1, 8}, "Single"], Bond[{1, 9}, "Single"], Bond[{2, 10}, "Single"], Bond[{2, 11}, "Single"], Bond[{3, 12}, "Single"], Bond[{4, 13}, "Single"], Bond[{4, 14}, "Single"], Bond[{5, 15}, "Single"], Bond[{5, 16}, "Single"], Bond[{6, 17}, "Single"], Bond[{7, 18}, "Single"], Bond[{7, 19}, "Single"]}, {}]]
Out[2]=

Compute the Xu index of a named entity:

In[3]:=
ResourceFunction["XuIndex"][Entity["Graph", "TutteGraph"]]
Out[3]=

Options (2) 

IncludeHydrogens (2) 

By default, hydrogens are ignored in the computation of a molecule Xu index:

In[4]:=
ResourceFunction["XuIndex"][Molecule["neopentane"]]
Out[4]=

Use IncludeHydrogens→All to account for hydrogens:

In[5]:=
ResourceFunction["XuIndex"][Molecule["neopentane"], IncludeHydrogens -> All]
Out[5]=

Applications (2) 

A list of alkanes:

In[6]:=
Short[alks = { Entity["Chemical", "Propane"], Entity["Chemical", "2Methylpropane"], Entity["Chemical", "Butane"], Entity["Chemical", "2,2Dimethylpropane"], Entity["Chemical", "2Methylbutane"], Entity["Chemical", "Pentane"], Entity["Chemical", "2,2Dimethylbutane"], Entity["Chemical", "2,3Dimethylbutane"], Entity["Chemical", "2Methylpentane"], Entity["Chemical", "3Methylpentane"], Entity["Chemical", "Hexane"], Entity["Chemical", "2,2,3Trimethylbutane"], Entity["Chemical", "2,2Dimethylpentane"], Entity["Chemical", "2,3Dimethylpentane"], Entity["Chemical", "2,4Dimethylpentane"], Entity["Chemical", "2Methylhexane"], Entity["Chemical", "3,3Dimethylpentane"], Entity["Chemical", "3Ethylpentane"], Entity["Chemical", "3Methylhexane"], Entity["Chemical", "Heptane"], Entity["Chemical", "Isoheptane"], Entity["Chemical", "2,2,3,3Tetramethylbutane"], Entity["Chemical", "2,2,3Trimethylpentane"], Entity["Chemical", "2,2,4Trimethylpentane"], Entity["Chemical", "2,2Dimethylhexane"], Entity["Chemical", "2,3,3Trimethylpentane"], Entity["Chemical", "2,3,4Trimethylpentane"], Entity["Chemical", "2,3Dimethylhexane"], Entity["Chemical", "2,4Dimethylhexane"], Entity["Chemical", "2,5Dimethylhexane"], Entity["Chemical", "2Methylheptane"], Entity["Chemical", "3,3Dimethylhexane"], Entity["Chemical", "3,4Dimethylhexane"], Entity["Chemical", "3Ethyl2Methylpentane"], Entity["Chemical", "3Ethyl3Methylpentane"], Entity["Chemical", "3Ethylhexane"], Entity["Chemical", "3Methylheptane"], Entity["Chemical", "4Methylheptane"], Entity["Chemical", "Octane"], Entity["Chemical", "2,2,3,3Tetramethylpentane"], Entity["Chemical", "2,2,3,4Tetramethylpentane"], Entity["Chemical", "2,2,3Trimethylhexane"], Entity["Chemical", "2,2,4,4Tetramethylpentane"], Entity["Chemical", "2,2,4Trimethylhexane"], Entity["Chemical", "2,2,5Trimethylhexane"], Entity["Chemical", "2,2Dimethylheptane"], Entity["Chemical", "2,3,3,4Tetramethylpentane"], Entity["Chemical", "2,3,3Trimethylhexane"], Entity["Chemical", "2,3,4Trimethylhexane"], Entity["Chemical", "2,3,5Trimethylhexane"], Entity["Chemical", "2,3Dimethylheptane"], Entity["Chemical", "2,4,4Trimethylhexane"], Entity["Chemical", "2,4Dimethyl3Ethylpentane"], Entity["Chemical", "2,4Dimethylheptane"], Entity["Chemical", "2,5Dimethylheptane"], Entity["Chemical", "2,6Dimethylheptane"], Entity["Chemical", "2Methyloctane"], Entity["Chemical", "3,3,4Trimethylhexane"], Entity["Chemical", "3,3Diethylpentane"], Entity["Chemical", "3,3Dimethylheptane"], Entity["Chemical", "3,4Dimethylheptane"], Entity["Chemical", "3,5Dimethylheptane"], Entity["Chemical", "3Ethyl2,2Dimethylpentane"], Entity["Chemical", "3Ethyl2,3Dimethylpentane"], Entity["Chemical", "3Ethyl2Methylhexane"], Entity["Chemical", "3Ethyl3Methylhexane"], Entity["Chemical", "3Ethyl4Methylhexane"], Entity["Chemical", "3Ethylheptane"], Entity["Chemical", "3Methyloctane"], Entity["Chemical", "4,4Dimethylheptane"], Entity["Chemical", "4Ethyl2Methylhexane"], Entity["Chemical", "4Ethylheptane"], Entity["Chemical", "4Methyloctane"], Entity["Chemical", "Isononane"], Entity["Chemical", "Nonane"], Entity["Chemical", "2,2,3,3Tetramethylhexane"], Entity["Chemical", "2,2,4,5TetramethylHexane"], Entity["Chemical", "2,2,4Trimethylheptane"], Entity["Chemical", "2,2,5,5Tetramethylhexane"], Entity["Chemical", "2,2Dimethyloctane"], Entity["Chemical", "2,3,4Trimethylheptane"], Entity["Chemical", "2,3,5Trimethylheptane"], Entity["Chemical", "2,3,6Trimethylheptane"], Entity["Chemical", "2,3Dimethyloctane"], Entity["Chemical", "2,4,4Trimethylheptane"], Entity["Chemical", "2,4,5TrimethylHeptane"], Entity["Chemical", "2,4,6Trimethylheptane"], Entity["Chemical", "2,4Dimethyloctane"], Entity["Chemical", "2,5,5Trimethylheptane"], Entity["Chemical", "2,5Dimethyloctane"], Entity["Chemical", "2,6Dimethyloctane"], Entity["Chemical", "2,7Dimethyloctane"], Entity["Chemical", "2Methylnonane"], Entity["Chemical", "3,3,4,4Tetramethylhexane"], Entity["Chemical", "3,3,4Trimethylheptane"], Entity["Chemical", "3,3,5Trimethylheptane"], Entity["Chemical", "3,3Dimethyloctane"], Entity["Chemical", "3,4,5Trimethylheptane"], Entity["Chemical", "3,4DiethylHexane"], Entity["Chemical", "3,4Dimethyloctane"], Entity["Chemical", "3,5Dimethyloctane"], Entity["Chemical", "3,6Dimethyloctane"], Entity["Chemical", "3Ethyl2Methylheptane"], Entity["Chemical", "3Ethyl3Methylheptane"], Entity["Chemical", "3Ethyl5Methylheptane"], Entity["Chemical", "3Ethyloctane"], Entity["Chemical", "3Methylnonane"], Entity["Chemical", "4,4DimethylOctane"], Entity["Chemical", "4,5Dimethyloctane"], Entity["Chemical", "4Ethyl2,2Dimethylhexane"], Entity["Chemical", "4Ethyloctane"], Entity["Chemical", "4Isopropylheptane"], Entity["Chemical", "4Methylnonane"], Entity["Chemical", "4Propylheptane"], Entity["Chemical", "5Ethyl2Methylheptane"], Entity["Chemical", "5Methylnonane"], Entity["Chemical", "Decane"], Entity["Chemical", "Isodecane"], Entity["Chemical", "2Methyldecane"], Entity["Chemical", "3Methyldecane"], Entity["Chemical", "4Methyldecane"], Entity["Chemical", "Undecane"], Entity["Chemical", "2,2,4,4,6PentamethylHeptane"], Entity["Chemical", "2,2,4,6,6Pentamethylheptane"], Entity["Chemical", "2Methylundecane"], Entity["Chemical", "Dodecane"], Entity["Chemical", "Isododecane"]}]
Out[6]=

Compute their Xu indices:

In[7]:=
xl = ResourceFunction["XuIndex"] /@ EntityValue[alks, "Molecule"];

Get their boiling points:

In[8]:=
bp = QuantityMagnitude[EntityValue[alks, "BoilingPoint"], "DegreesCelsius"];

The Xu indices show a strong correlation with the boiling points of alkanes. Visualize the trend:

In[9]:=
da = Transpose[{xl, bp}]; ListPlot[da, Epilog -> {Directive[ AbsoluteThickness[2], RGBColor[0.368417, 0.506779, 0.709798]], ResourceFunction["TrendLine"][da]}, {Axes -> None, Frame -> True}]
Out[10]=

Compute the correlation coefficient:

In[11]:=
Correlation[xl, bp]
Out[11]=

Generate all alkanes with 7 carbon atoms (heptanes) using the resource function AlkaneIsomers:

In[12]:=
heptanes = ResourceFunction["AlkaneIsomers"][7]
Out[12]=

Sort the heptane isomers by their Xu index. This effectively sorts them from "most branched" to "least branched":

In[13]:=
SortBy[heptanes, ResourceFunction["XuIndex"]]
Out[13]=

Requirements

Wolfram Language 12.3 (May 2021) or above

Version History

  • 1.0.0 – 18 December 2023

Source Metadata

  • Citation:
    • Ren B., "A New Topological Index for QSPR of Alkanes." Journal of Chemical Information and Computer Sciences, vol. 39, no. 1, 139–143, 1999.
      DOI: 10.1021/ci980098p

Related Resources

Author Notes

XuIndex currently does not implement Ren's modification for molecules with multiple bonds and/or heteroatoms; this extension is reserved for a future update.

License Information