# bidirectedEdgesMatrix -- matrix corresponding to the bidirected edges of a bigraph or a mixed graph

## Synopsis

• Usage:
bidirectedEdgesMatrix R
• Inputs:
• R, a ring, which should be a gaussianRing created with an instance of the type Bigraph or an instance of the type MixedGraph
• Outputs:
• , the $n \times{} n$ covariance matrix of the noise variables in the Gaussian graphical model of a mixed graph.

## Description

This method returns the $n \times{} n$ covariance matrix of the noise variables in the Gaussian graphical model. The diagonal in this matrix consists of the indeterminates $p_{(i,i)}$. Each off-diagonal entry is zero unless there is a bidirected edge between i and j in which case the corresponding entry in the matrix is the indeterminate $p_{(i,j)}$. The documentation of gaussianRing further describes the indeterminates $p_{(i,j)}$.

 i1 : G = mixedGraph(digraph {{b,{c,d}},{c,{d}}},bigraph {{a,d}}) o1 = MixedGraph{Bigraph => Bigraph{a => {d}} } d => {a} Digraph => Digraph{b => {c, d}} c => {d} d => {} Graph => Graph{} o1 : MixedGraph i2 : R = gaussianRing G o2 = R o2 : PolynomialRing i3 : compactMatrixForm =false; i4 : bidirectedEdgesMatrix R o4 = | p 0 0 p | | a,a a,d | | | | 0 p 0 0 | | b,b | | | | 0 0 p 0 | | c,c | | | | p 0 0 p | | a,d d,d | 4 4 o4 : Matrix R <--- R

For mixed graphs that also have undirected edges, the size of the matrix coincides with the number of elements in compW, which depends on the vertex partition built in partitionLMG.

 i5 : G = mixedGraph(digraph {{1,3},{2,4}},bigraph {{3,4}},graph {{1,2}}); i6 : R = gaussianRing G o6 = R o6 : PolynomialRing i7 : bidirectedEdgesMatrix R o7 = | p p | | 3,3 3,4 | | | | p p | | 3,4 4,4 | 2 2 o7 : Matrix R <--- R

Bidirected graphs can also be considered:

 i8 : G = bigraph {{a,d},{b},{c}} o8 = Bigraph{a => {d}} b => {} c => {} d => {a} o8 : Bigraph i9 : R = gaussianRing G o9 = R o9 : PolynomialRing i10 : bidirectedEdgesMatrix R o10 = | p 0 0 p | | a,a a,d | | | | 0 p 0 0 | | b,b | | | | 0 0 p 0 | | c,c | | | | p 0 0 p | | a,d d,d | 4 4 o10 : Matrix R <--- R