Entering a (graded bipartite) graph

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What follows is a description of the functions that let one enter a general (graded bipartite) graph and easily enter common graphs (like the A-D-E Dynkin diagrams).

Functions

In[1]:= <<FusionAtlas`

Loading FusionAtlas` version 0 Read more at http://tqft.net/wiki/Atlas_of_subfactors

In[2]:= ?GradedBigraph A GradedBigraph is a list of matrices which represents a bipartite graph whose vertices are graded by distance from a chosen first vertex ('depth'). The nth matrix in the list is the adjacency matrix for the depth n vertices (which label columns) and the depth n+1 vertices (which label rows).

So, for example,

In[3]:=	GradedBigraph[{{1}},{{1}},{{1},{1}},{{1,0},{0,1}}]
Out[3]=	GradedBigraph[{{1}}, {{1}}, {{1}, {1}}, {{1, 0}, {0, 1}}]

In[4]:= ?DisplayBigraph DisplayBigraph[g] produces a Graphics object representing the GradedBigraph g.

DisplayBigraph produces output of the following form:

In[6]:=	DisplayBigraph[GradedBigraph[{{1}},{{1}},{{1},{1}},{{1,0},{0,1}}]]
Out[6]=	-Graphics-

In[7]:= ?AnBigraph AnBigraph[n] returns the bigraph form of the Dynkin diagram A_n

In[8]:= ?DnBigraph DnBigraph[n] returns the bigraph form of the Dynkin diagram D_n

In[9]:= ?EnBigraph EnBigraph[n] returns the bigraph form of the Dynkin diagram E_n

In[10]:= ?trivalentBigraph trivalentBigraph[i,j,k] returns a bigraph with one trivalent vertex and three legs, the legs having i, j, and k vertices

In[11]:= ?haagerupFamilyBigraph haagerupFamilyBigraph[n] returns trivalentBigraph[n,3,3], ie the bigraph form of a graph with one trivalent vertex and three legs, the legs having n, 3, and 3 vertices, ie o-o-...-o<8=8=8 .

In[12]:= ?dualHaagerupFamilyBigraph dualHaagerupFamilyBigraph[n] returns the bigraph form of the graph which has two adjacent trivalent vertices and four legs, the legs having n, 1, 1, and 1 vertices, ie __...__!_!_

In[13]:= ?HaagerupBigraph HaagerupBigraph is the bigraph form of the graph o-o-o-o<8=8=8, the principal graph of the Haagerup subfactor constructed by Asaeda and Haagerup in 'Exotic Subfactors of Finite Depth ....'

Here's a better picture of the Haagerup graph:

In[15]:=	DisplayBigraph[HaagerupBigraph]
Out[15]=	-Graphics-

In[16]:= ?DualHaagerupBigraph HaagerupBigraph is the bigraph form of the graph ___!_!_, the dual principal graph of the Haagerup subfactor constructed by Asaeda and Haagerup in 'Exotic Subfactors of Finite Depth ....'

Here's a better picture of the dual Haagerup graph:

In[18]:=	DisplayBigraph[DualHaagerupBigraph]
Out[18]=	-Graphics-

Miscellany

In[19]:= ?GraphToString BigraphToString[g_] returns a string suitable for use both as a filename and as a Mathematica symbol name, encoding the graph g.