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# graphic(List,List,PolynomialRing) -- make a graphic arrangement

## Synopsis

• Function: graphic
• Usage:
graphic(E, V, R)
graphic(E, R)
graphic(E, V)
graphic E
• Inputs:
• E, a list, the edges of a graph expressed as a list of pairs of vertices as specified in $V$
• V, a list, the vertices of a graph expressed as a list of elements
• R, , an optional coordinate ring for the arrangement or a ring to be interpreted as a coefficient ring
• Outputs:
• , associated to the given graph

## Description

A graph $G$ is specified by a list $V$ of vertices and a list $E$ of pairs of vertices. When $V$ is not specified, it is assumed to be the list $1, 2, \ldots, n$, where $n$ is the largest integer appearing as a vertex of $E$. The graphic arrangement $A(G)$ of $G$ is the subarrangement of the type $A_{n-1}$ arrangement with hyperplanes $x_i-x_j$ for each edge $\{i,j\}$ of the graph $G$.

 i1 : G = {{1,2},{2,3},{3,4},{4,1}}; -- a four-cycle i2 : AG = graphic G o2 = {- x + x , - x + x , - x + x , x - x } 1 2 2 3 3 4 1 4 o2 : Hyperplane Arrangement  i3 : rank AG -- the number of vertices minus number of components o3 = 3 i4 : ring AG o4 = QQ[x ..x ] 1 4 o4 : PolynomialRing

One can also specify the ambient ring.

 i5 : AG' = graphic(G,QQ[x,y,z,w]) -- four variables because there are 4 vertices o5 = {- x + y, - y + z, - z + w, x - w} o5 : Hyperplane Arrangement  i6 : ring AG' o6 = QQ[x..z, w] o6 : PolynomialRing

Occasionally, one might want to give labels to the vertices. These labels can be anything!

 i7 : V = {"a", "b", "c", "d"}; i8 : E = {{"a","b"}, {"b", "c"}, {"c","d"}, {"d","a"}}; i9 : graphic(E, V) o9 = {- x + x , - x + x , - x + x , x - x } 1 2 2 3 3 4 1 4 o9 : Hyperplane Arrangement 

The vertices can also be the variables of a polynomial ring.

 i10 : R = QQ[a,b,c,d]; i11 : arr = graphic({{a,b},{b,c},{c,d},{d,a}}, gens R, R) o11 = {- a + b, - b + c, - c + d, a - d} o11 : Hyperplane Arrangement  i12 : ring arr === R o12 = true

Loops and parallel edges are allowed.

 i13 : graphic({{1,2}, {1,2}}) o13 = {- x + x , - x + x } 1 2 1 2 o13 : Hyperplane Arrangement  i14 : graphic({{1,1}, {1,2}}) o14 = {0, - x + x } 1 2 o14 : Hyperplane Arrangement