# graphic -- Make a graphic arrangement

## Synopsis

• Usage:
graphic(G) or graphic(G,k) or graphic(G,R)
• Inputs:
• G, a list, a graph, expressed as a list of pairs of vertices
• k, a ring, an optional coefficient ring, by default QQ
• R, , an optional coordinate ring for the arrangement
• Outputs:
• , the graphic arrangement from graph G

## Description

A graph G has vertices 1, 2, ..., n, and its edges are a list of lists of length 2. The graphic arrangement A(G) of G is, by definition, the subarrangement of the type A_(n-1) arrangement with hyperplanes x_i-x_j for each edge {i,j} of 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 : describe AG o3 = {- x + x , - x + x , - x + x , x - x } 1 2 2 3 3 4 1 4 i4 : rank AG -- the number of vertices minus number of components o4 = 3 i5 : ring AG o5 = QQ[x ..x ] 1 4 o5 : PolynomialRing i6 : ring graphic(G,ZZ[x,y,z,w]) o6 = ZZ[x..z, w] o6 : PolynomialRing