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Instant-use add-on functions for the Wolfram Language
Function Repository Resource:
Wiener
Compute the Wiener index of a graph
Contributed by:
Jon McLoone
ResourceFunction["WienerIndex"][g] computes the Wiener index of the graph g. |
Details and Options
The Wiener index is defined to be the sum of the lengths of the shortest paths between all pairs of vertices in the given graph.
In chemistry, the Wiener index is typically applied to the hydrogen-depleted molecule graph where it has been shown to be correlated to various chemical properties.
Examples
Basic Examples (3)
The Wiener index of a simple graph:
| In[1]:= | ![ResourceFunction["WienerIndex"][\!\(\*
GraphicsBox[
NamespaceBox["NetworkGraphics",
DynamicModuleBox[{Typeset`graph = HoldComplete[
Graph[{1, 2, 3, 4}, {Null, {{1, 2}, {1, 3}, {1, 4}}}, {GraphLayout -> "StarEmbedding"}]]},
TagBox[GraphicsGroupBox[
GraphicsComplexBox[{{0., 0.}, {
0.8660254037844389, -0.5000000000000012}, {
1.8369701987210297`*^-16, 1.}, {-0.8660254037844386, -0.49999999999999917`}}, {
{Hue[0.6, 0.7, 0.5], Opacity[0.7], Arrowheads[0.], ArrowBox[{{1, 2}, {1, 3}, {1, 4}}, 0.020399597244776413`]},
{Hue[0.6, 0.2, 0.8], EdgeForm[{GrayLevel[0], Opacity[0.7]}], DiskBox[1, 0.020399597244776413], DiskBox[2, 0.020399597244776413], DiskBox[3, 0.020399597244776413], DiskBox[4, 0.020399597244776413]}}]],
MouseAppearanceTag["NetworkGraphics"]],
AllowKernelInitialization->False]],
DefaultBaseStyle->"NetworkGraphics",
FormatType->TraditionalForm,
FrameTicks->None]\)]](https://www.wolframcloud.com/obj/resourcesystem/images/9ba/9ba8469e-bb44-4328-b404-e0a9944b880a/75e2160547e10ec3.png) |
| Out[1]= |  |
More densely connected graphs have lower WienerIndex values:
| In[2]:= | ![ResourceFunction["WienerIndex"][\!\(\*
GraphicsBox[
NamespaceBox["NetworkGraphics",
DynamicModuleBox[{Typeset`graph = HoldComplete[
Graph[{1, 2, 3, 4}, {Null,
SparseArray[
Automatic, {4, 4}, 0, {1, {{0, 3, 6, 9, 12}, {{2}, {3}, {4}, {1}, {3}, {4}, {
1}, {2}, {4}, {1}, {2}, {3}}}, Pattern}]}, {GraphLayout -> "StarEmbedding"}]]},
TagBox[GraphicsGroupBox[
GraphicsComplexBox[{{0., 0.}, {
0.8660254037844389, -0.5000000000000012}, {
1.8369701987210297`*^-16, 1.}, {-0.8660254037844386, -0.49999999999999917`}}, {
{Hue[0.6, 0.7, 0.5], Opacity[0.7], Arrowheads[0.], ArrowBox[{{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}, 0.020399597244776413`]},
{Hue[0.6, 0.2, 0.8], EdgeForm[{GrayLevel[0], Opacity[0.7]}], DiskBox[1, 0.020399597244776413], DiskBox[2, 0.020399597244776413], DiskBox[3, 0.020399597244776413], DiskBox[4, 0.020399597244776413]}}]],
MouseAppearanceTag["NetworkGraphics"]],
AllowKernelInitialization->False]],
DefaultBaseStyle->"NetworkGraphics",
FormatType->TraditionalForm,
FrameTicks->None]\)]](https://www.wolframcloud.com/obj/resourcesystem/images/9ba/9ba8469e-bb44-4328-b404-e0a9944b880a/504c79f4df17ae53.png) |
| Out[2]= |  |
| In[3]:= | ![ResourceFunction["WienerIndex"][\!\(\*
GraphicsBox[
NamespaceBox["NetworkGraphics",
DynamicModuleBox[{Typeset`graph = HoldComplete[
Graph[{1, 2, 3, 4}, {Null, {{1, 2}, {2, 3}, {3, 4}}}, {VertexCoordinates -> {{1, 1}, {2, 2}, {3, 1}, {4, 2}}}]]},
TagBox[GraphicsGroupBox[
GraphicsComplexBox[{{1., 1.}, {2., 2.}, {3., 1.}, {4., 2.}}, {
{Hue[0.6, 0.7, 0.5], Opacity[0.7], Arrowheads[0.], ArrowBox[{{1, 2}, {2, 3}, {3, 4}}, 0.030239520958083826`]},
{Hue[0.6, 0.2, 0.8], EdgeForm[{GrayLevel[0], Opacity[0.7]}], DiskBox[1, 0.030239520958083826], DiskBox[2, 0.030239520958083826], DiskBox[3, 0.030239520958083826], DiskBox[4, 0.030239520958083826]}}]],
MouseAppearanceTag["NetworkGraphics"]],
AllowKernelInitialization->False]],
DefaultBaseStyle->"NetworkGraphics",
FormatType->TraditionalForm,
FrameTicks->None]\)]](https://www.wolframcloud.com/obj/resourcesystem/images/9ba/9ba8469e-bb44-4328-b404-e0a9944b880a/090bbee2a2f55e5a.png){kind=small} |
| Out[3]= |  |
By default, the WienerIndex of a molecule is the WienerIndex of the hydrogen-depleted molecule graph. That is, the molecule graph with all hydrogen atoms removed:
| In[4]:= | ![ResourceFunction["WienerIndex"][
Molecule[{"C", "C", "C", "C", "H", "H", "H", "H", "H", "H", "H", "H", "H", "H"}, {
Bond[{1, 2}, "Single"],
Bond[{1, 3}, "Single"],
Bond[{1, 4}, "Single"],
Bond[{1, 5}, "Single"],
Bond[{2, 6}, "Single"],
Bond[{2, 7}, "Single"],
Bond[{2, 8}, "Single"],
Bond[{3, 9}, "Single"],
Bond[{3, 10}, "Single"],
Bond[{3, 11}, "Single"],
Bond[{4, 12}, "Single"],
Bond[{4, 13}, "Single"],
Bond[{4, 14}, "Single"]}, {AtomCoordinates -> QuantityArray[
StructuredArray`StructuredData[{14, 3}, {CompressedData["
1:eJxTTMoPSmViYGDgA2JmBkwQ/kZ/tzr/9f1bT5Ttmy/1cP/azHuFXX1f7U8+
tV9yf98++9yj/zZVf/pmv/ZXDJB5aj9MXP1Q2/LwU2/2nwRp0/8EF0c3v4/t
g5hHwPf9H5PPxHqfeLi/TtYi3SXzm/3UK5wZSjmf7Ws+bQjInrXd/qtXZJtF
GKMDhH9+v0fAH4ni6z/2v2IxETSz+WEPE3dMeHpB6fZ3++cg6/jO7GdvnOrc
DTRnP98c40VbGBwg7npk/7kvuERl+vn9F11ufPji9dd+8sqmQM+5n/bDzCnd
Kvr79Lu3+z85nk+7+vzTfph7OsEWPNxvDmYwHICp/xn8eOnsI//2PwPb+wFu
DgBbdbEe
"], "Angstroms", {{1}, {2}}}]], AtomDiagramCoordinates -> {{2.865999937057495, 0.25}, {
3.7320001125335693`, 0.75}, {2., 0.75}, {
2.865999937057495, -0.75}, {2.865999937057495, 0.8700000047683716}, {4.041999816894531, 0.21310000121593475`}, {
4.269000053405762, 1.059999942779541}, {3.421999931335449, 1.2869000434875488`}, {2.309999942779541, 1.2869000434875488`}, {
1.4630999565124512`, 1.059999942779541}, {1.690000057220459, 0.21310000121593475`}, {2.246000051498413, -0.75}, {
2.865999937057495, -1.3700000047683716`}, {
3.4860000610351562`, -0.75}}}]]](https://www.wolframcloud.com/obj/resourcesystem/images/9ba/9ba8469e-bb44-4328-b404-e0a9944b880a/66e91dcde318f122.png) |
| Out[4]= |  |
A different isomer with the same atoms can have a different WienerIndex:
| In[5]:= | ![ResourceFunction["WienerIndex"][
Molecule[{"C", "C", "C", "C", "H", "H", "H", "H", "H", "H", "H", "H", "H", "H"}, {
Bond[{1, 2}, "Single"],
Bond[{1, 3}, "Single"],
Bond[{1, 5}, "Single"],
Bond[{1, 6}, "Single"],
Bond[{2, 4}, "Single"],
Bond[{2, 7}, "Single"],
Bond[{2, 8}, "Single"],
Bond[{3, 9}, "Single"],
Bond[{3, 10}, "Single"],
Bond[{3, 11}, "Single"],
Bond[{4, 12}, "Single"],
Bond[{4, 13}, "Single"],
Bond[{4, 14}, "Single"]}, {AtomCoordinates -> QuantityArray[
StructuredArray`StructuredData[{14, 3}, {CompressedData["
1:eJwBYQGe/iFib1JlAgAAAA4AAAADAAAAyXGndLD+2D9VNNb+zvbRv29JDtjV
5OK/yqFFtvP92L8hsd09QPfRP+Z4BaIn5eI/dqvnpPcN/D/jj6LO3EPZP/yP
TIdOT+W/dqvnpPcN/L9INlfNc0TZvzsZHCWvTuU/9DP1ukXg4D/VYBqGjwj2
v7cr9MEyNt2/BRTq6SPwxb8kjFU5o5y2vzZeukkMgvi/DECjdOnf4L8prir7
rgj2P9JyoIfaNt0/dbUV+8vuxT+NzMjFvZ62P8SZX80Bgvg/gJpattaXAkDa
l8f1SrSPv5+rrdhfdvi/PtAKDFnd+T/qymd5Htz3P8nJxK2CmOm/hEcbR6yF
AkC/vtalRujJPwg8MIDwodE/gJpattaXAsAuyzIfa6GPPxKI1/ULdvg/BOJ1
/YLd+b+WfVcE/9v3v7hYUYNpmOk/TOXtCKeFAsBy/5Hp0OnJv/oq+dhdoNG/
KH7CaA==
"], "Angstroms", {{1}, {2}}}]], AtomDiagramCoordinates -> {{2.865999937057495, -0.25}, {
3.732100009918213, 0.25}, {2., 0.25}, {
4.598100185394287, -0.25}, {
2.4674999713897705`, -0.7249000072479248}, {
3.2646000385284424`, -0.7249000072479248}, {4.1305999755859375`, 0.7249000072479248}, {3.3334999084472656`, 0.7249000072479248}, {
2.309999942779541, 0.786899983882904}, {1.4630999565124512`, 0.5600000023841858}, {1.690000057220459, -0.28690001368522644`}, {
4.288099765777588, -0.786899983882904}, {
5.135000228881836, -0.5600000023841858}, {4.908100128173828, 0.28690001368522644`}}}]]](https://www.wolframcloud.com/obj/resourcesystem/images/9ba/9ba8469e-bb44-4328-b404-e0a9944b880a/562373d5ec84885a.png) |
| Out[5]= |  |
WienerIndex can accept "Graph" or "Chemical" entities:
| In[6]:= | ![ResourceFunction["WienerIndex"][Entity["Chemical", "Butane"]]](https://www.wolframcloud.com/obj/resourcesystem/images/9ba/9ba8469e-bb44-4328-b404-e0a9944b880a/71626f0cf4ec1a70.png){kind=small} |
| Out[6]= |  |
| In[7]:= | ![ResourceFunction["WienerIndex"][Entity["Graph", {"Cycle", 20}]]](https://www.wolframcloud.com/obj/resourcesystem/images/9ba/9ba8469e-bb44-4328-b404-e0a9944b880a/64080f50e10ca6af.png){kind=small} |
| Out[7]= |  |
Options (2)
IncludeHydrogens (2)
WienerIndex of a molecule ignores hydrogen atoms:
| In[8]:= | ![ResourceFunction["WienerIndex"][Entity["Chemical", "Butane"]]](https://www.wolframcloud.com/obj/resourcesystem/images/9ba/9ba8469e-bb44-4328-b404-e0a9944b880a/4846ada8e47897ff.png){kind=small} |
| Out[8]= |  |
The option IncludeHydrogens→ All will have them taken into account:
| In[9]:= | ![ResourceFunction["WienerIndex"][Entity["Chemical", "Butane"], IncludeHydrogens -> All]](https://www.wolframcloud.com/obj/resourcesystem/images/9ba/9ba8469e-bb44-4328-b404-e0a9944b880a/4be6c62642673602.png){kind=small} |
| Out[9]= |  |
Applications (2)
Create OEIS sequence A292054 - the Wiener index of n×n knight's tour graphs:
| In[10]:= | ![Table[ResourceFunction["WienerIndex"][KnightTourGraph[n, n]], {n, 20}]](https://www.wolframcloud.com/obj/resourcesystem/images/9ba/9ba8469e-bb44-4328-b404-e0a9944b880a/17103539a0535181.png){kind=small} |
| Out[10]= |  |
Create OEIS sequence A034828 - the Wiener index of n-cycle graphs:
| In[11]:= | ![Table[ResourceFunction["WienerIndex"][CycleGraph[n]], {n, 20}]](https://www.wolframcloud.com/obj/resourcesystem/images/9ba/9ba8469e-bb44-4328-b404-e0a9944b880a/263c18b10dc1c1ba.png){kind=small} |
| Out[11]= |  |
Generate all alkanes with 7 carbon atoms (heptanes) using the resource function AlkaneIsomers:
| In[12]:= | ![heptanes = ResourceFunction["AlkaneIsomers"][7]](https://www.wolframcloud.com/obj/resourcesystem/images/9ba/9ba8469e-bb44-4328-b404-e0a9944b880a/5e3a95cc817e3543.png){kind=small} |
| Out[12]= |  |
Sort the heptane isomers by their Wiener index. This effectively sorts them from "most branched" to "least branched":
| In[13]:= | ![SortBy[heptanes, ResourceFunction["WienerIndex"]]](https://www.wolframcloud.com/obj/resourcesystem/images/9ba/9ba8469e-bb44-4328-b404-e0a9944b880a/26469aad77ea4ad6.png){kind=small} |
| Out[13]= |  |
Properties and Relations (2)
For an acyclic graph the Wiener index is the same as the Szeged index:
| In[14]:= | ![ResourceFunction["WienerIndex"][ \!\(\*
GraphicsBox[
NamespaceBox["NetworkGraphics",
DynamicModuleBox[{Typeset`graph = HoldComplete[
Graph[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {Null, {{3, 4}, {5, 6}, {5, 3}, {7, 8}, {7, 1}, {7, 2}, {
7, 3}, {9, 10}, {9, 2}}}]]},
TagBox[GraphicsGroupBox[
GraphicsComplexBox[{{0., 1.466033332275663}, {
0.7330166661378313, 1.466033332275663}, {1.8325416653445783`,
1.466033332275663}, {1.4660333322756627`, 0.7330166661378315}, {2.199049998413494, 0.7330166661378315}, {2.199049998413494, 0.}, {
1.2827791657412049`, 2.199049998413494}, {
2.5655583314824097`, 1.466033332275663}, {0.7330166661378313,
0.7330166661378315}, {0.7330166661378313, 0.}}, {
{Hue[0.6, 0.7, 0.5], Opacity[0.7], Arrowheads[0.], ArrowBox[{{1, 7}, {2, 7}, {2, 9}, {3, 4}, {3, 5}, {3, 7}, {
5, 6}, {7, 8}, {9, 10}}, 0.027040802458717428`]},
{Hue[0.6, 0.2, 0.8], EdgeForm[{GrayLevel[0], Opacity[0.7]}], DiskBox[1, 0.027040802458717428], DiskBox[2, 0.027040802458717428], DiskBox[3, 0.027040802458717428], DiskBox[4, 0.027040802458717428], DiskBox[5, 0.027040802458717428], DiskBox[6, 0.027040802458717428], DiskBox[7, 0.027040802458717428], DiskBox[8, 0.027040802458717428], DiskBox[9, 0.027040802458717428], DiskBox[10, 0.027040802458717428]}}]],
MouseAppearanceTag["NetworkGraphics"]],
AllowKernelInitialization->False]],
DefaultBaseStyle->"NetworkGraphics",
FormatType->TraditionalForm,
FrameTicks->None]\)]](https://www.wolframcloud.com/obj/resourcesystem/images/9ba/9ba8469e-bb44-4328-b404-e0a9944b880a/3df6d0420cdae96c.png) |
| Out[14]= |  |
| In[15]:= | ![ResourceFunction["SzegedIndex"][\!\(\*
GraphicsBox[
NamespaceBox["NetworkGraphics",
DynamicModuleBox[{Typeset`graph = HoldComplete[
Graph[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {Null, {{3, 4}, {5, 6}, {5, 3}, {7, 8}, {7, 1}, {7, 2}, {
7, 3}, {9, 10}, {9, 2}}}]]},
TagBox[GraphicsGroupBox[
GraphicsComplexBox[{{0., 1.466033332275663}, {
0.7330166661378313, 1.466033332275663}, {1.8325416653445783`,
1.466033332275663}, {1.4660333322756627`, 0.7330166661378315}, {2.199049998413494, 0.7330166661378315}, {2.199049998413494, 0.}, {
1.2827791657412049`, 2.199049998413494}, {
2.5655583314824097`, 1.466033332275663}, {0.7330166661378313,
0.7330166661378315}, {0.7330166661378313, 0.}}, {
{Hue[0.6, 0.7, 0.5], Opacity[0.7], Arrowheads[0.], ArrowBox[{{1, 7}, {2, 7}, {2, 9}, {3, 4}, {3, 5}, {3, 7}, {
5, 6}, {7, 8}, {9, 10}}, 0.027040802458717428`]},
{Hue[0.6, 0.2, 0.8], EdgeForm[{GrayLevel[0], Opacity[0.7]}], DiskBox[1, 0.027040802458717428], DiskBox[2, 0.027040802458717428], DiskBox[3, 0.027040802458717428], DiskBox[4, 0.027040802458717428], DiskBox[5, 0.027040802458717428], DiskBox[6, 0.027040802458717428], DiskBox[7, 0.027040802458717428], DiskBox[8, 0.027040802458717428], DiskBox[9, 0.027040802458717428], DiskBox[10, 0.027040802458717428]}}]],
MouseAppearanceTag["NetworkGraphics"]],
AllowKernelInitialization->False]],
DefaultBaseStyle->"NetworkGraphics",
FormatType->TraditionalForm,
FrameTicks->None]\)]](https://www.wolframcloud.com/obj/resourcesystem/images/9ba/9ba8469e-bb44-4328-b404-e0a9944b880a/267e25a2f784578b.png) |
| Out[15]= |  |
GraphData provides the pre-computed WienerIndex for many special graphs:
| In[16]:= | ![GraphData["PetersenGraph", "WienerIndex"]](https://www.wolframcloud.com/obj/resourcesystem/images/9ba/9ba8469e-bb44-4328-b404-e0a9944b880a/224c555d66c47d5d.png){kind=small} |
| Out[16]= |  |
We can confirm this number by computing it on the graph:
| In[17]:= | ![ResourceFunction["WienerIndex"][GraphData["PetersenGraph"]]](https://www.wolframcloud.com/obj/resourcesystem/images/9ba/9ba8469e-bb44-4328-b404-e0a9944b880a/061a41a56c2d02a5.png){kind=small} |
| Out[17]= |  |
Publisher
Related Links
Requirements
Wolfram Language 12.3 (May 2021) or above
Version History
- 1.0.0 – 11 March 2024
Source Metadata
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License Information
This work is licensed under a Creative Commons Attribution 4.0 International License