Computing properties of graphs

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After entering a (graded bipartite) graph, one can compute many properties of the graph (like the Perron-Frobenius dimensions of its vertices) and find graphs which extend a given graph, using the following functions.

Functions

In[1]:= ?RankAtDepth RankAtDepth[g,n] gives the number of vertices at depth n of the bipartite graph g

In[2]:= ?GraphDepth GraphDepth[g] gives the maximum depth of the bipartite graph g

In[3]:= ?GraphRank GraphRank[g] gives the rank (ie the number of vertices) of the bipartite graph g

In[4]:= ?GraphEvenRank GraphRank[g] gives the even rank (ie the number of even vertices) of the bipartite graph g

In[5]:= ?GraphOddRank GraphRank[g] gives the odd rank (ie the number of odd vertices) of the bipartite graph g

In[6]:= ?GraphAdjacencyMatrix GraphAdjacencyMatrix[g] gives the adjacency matrix of the graph g

In[7]:= ?DimensionOfGenerator DimensionOfGenerator[g] gives the Perron-Frobenius dimension of the unique vertex at depth 1 of the bipartite graph g (assuming there is such a vertex)

In[8]:= ?NumericDimensionOfGenerator NumericDimensionOfGenerator[g] gives the numeric value of the Perron-Frobenius dimension of the unique vertex at depth 1 of the bipartite graph g (assuming there is such a vertex)

In[9]:= ?GraphIndex GraphIndex[g] gives the square of the Perron-Frobenius eigenvalue of the adjacency matrix ofthe bipartite graph g

In[10]:= ?DimensionAtMostQ DimensionAtMostQ[x][g] returns True if the dimension (of the generator) of the bipartite graph g is less than x. DimensionAtMostQ[g1][g2] returns True if the dimension (of the generator) of the bipartite graph g1 is more than the dimension (of the generator) of the bipartite graph g2.

In[11]:= ?DimensionsByDepth DimensionsByDepth[g] gives a list of the Perron-Frobenius dimensions of the vertices of the bipartite graph g, sorted by depth.

In[12]:= ?NumericDimensionsByDepth NumericDimensionsByDepth[g] gives a list of the numeric values of the Perron-Frobenius dimensions of the vertices of the bipartite graph g, sorted by depth

In[13]:= ?DimensionOfLowWeightSpace DimensionOfLowWeightSpace[g,r] gives the dimension of the subspace of P_r which is orthogonal to the lift (ie, annular consequences) of the elements of P_{r-1}, where P is a planar algebra with principal graph g. This is the same as the number of irreducible modules of lowest weight r in the decomposition of P into annular Temperley-Lieb modules.

In[14]:= ?AnnularTanglesSubgraphTest AnnularTanglesSubgraphTest[g] applies the annular tangles test (described on p.33 of Vaughan Jones' 'Annular Structure of Subfactors') up to the depth of the graph g, and can rule out g from being the initial segment of any principal graph

In[15]:= ?AnnularTanglesTest AnnularTanglesTest[g_] checks for the graph g the condition on a principal graph described on p.33 of Vaughan Jones' 'Annular Structure of Subfactors'.

In[16]:= ?FindBigraphExtensions FindBigraphExtensions[c][b,r] gives all bipartite graphs which look like b up to the depth of b, have r new vertices at one plus the depth, and have index less than or equal to c. Currently, FindBigraphExtensions only returns graphs g for which AnnularTanglesSubgraphTest[g] returns True.

In[17]:= ?FindBigraphExtensionsUpToRank FindBigraphExtensionsUpToRank[c][b,r] gives all bipartite graphs which look like b up to the depth of b, have at most r new vertices at one plus the depth, and have index less than or equal to c. It can also be run on a list of bipartite graphs. Currently, FindBigraphExtensions only returns graphs g for which AnnularTanglesSubgraphTest[g] returns True.

In[18]:= ?FindBigraphExtensionsUpToRankAndDepth FindBigraphExtensionsUpToRankAndDepth[c][b,r,d] gives all bipartite graphs which look like b up to the depth of b, have at most r new vertices between the depth of b and depth d, and have index less than or equal to c. Currently, FindBigraphExtensions only returns graphs g for which AnnularTanglesSubgraphTest[g] returns True.