Editing Cuboctahedron (section)

== Graph ==
The skeleton of a cuboctahedron may be represented as the [[Graph (discrete mathematics)|graph]], one of the [[Archimedean graph]].  It has 12 [[Vertex (graph theory)|vertices]] and 24 edges. It is [[quartic graph]], which is four vertices connecting each vertex.{{sfn|Read|Wilson|1998|p=269}}

The graph of a cuboctahedron may be constructed as the [[line graph]] of the [[cubical graph]], making it becomes the [[locally linear graph]].{{sfn|Fan|1996}}

The 24 edges can be partitioned into 2 sets isomorphic to tetrahedral symmetry. The edges can also be partitioned into 4 hexagonal cycles, representing [[centrosymmetry]], with only opposite vertices and edges in the same transitivity class.{{citation needed|date=March 2025}}
{| class=wikitable
!colspan=2|Octahedral (48 [[automorphism]])||colspan=2|Tetrahedral (24 aut)||Centrosymmetric (2 aut)
|-
|[[File:Cuboctahedron-graph-colored.svg|160px]]
|
{| class=wikitable
|+ Configuration
|- 
|\||style="background-color:#000000; color: #E0E0E0"|v<sub>1</sub>||style="background-color:#FF0000;"|e<sub>1</sub>
|- align=right valign=top
|style="background-color:#000000; color:#E0E0E0"|v<sub>1</sub>||style="background-color:#E0F0FF"|12|||4
|- align=right
|style="background-color:#FF0000;"|e<sub>1</sub>|||2||style="background-color:#f0FFE0"|24
|}
|[[File:Cuboctahedron-graph-tetrahedral-colored.svg|160px]]
|
{| class=wikitable
|+ Configuration
|-
|\||style="background-color:#000000; color: #E0E0E0"|v<sub>1</sub>||style="background-color:#3CB44B;"|e<sub>1</sub>||style="background-color:#E6194B;"|e<sub>2</sub>
|- align=right
|style="background-color:#000000; color:#E0E0E0"|v<sub>1</sub>||style="background-color:#E0F0FF"|12|||2|||2
|- align=right
|style="background-color:#3CB44B;"|e<sub>1</sub>|||2||style="background-color:#f0FFE0"|12||style="background-color:#f0FFE0"|*
|- align=right
|style="background-color:#E6194B;"|e<sub>2</sub>|||2||style="background-color:#f0FFE0"|*||style="background-color:#f0FFE0"|12
|}
|[[File:Cuboctahedral_graph-ortho-4color.svg|160px]]

|}