undirectedEdgesMatrix -- matrix corresponding to the edges of an undirected graph

Synopsis

• Usage:
undirectedEdgesMatrix(R)
• Inputs:
• Outputs:
• , the n x n symmetric concentration matrix of an undirected gaussian graphical model.

Description

This symmetric matrix has entries $k_{(i,i)}$ along the diagonal and entry $k_{(i,j)}$ in the $(i,j)$ position if there is an edge between i and j, and a zero otherwise. The documentation of gaussianRing further describes the indeterminates $k_{(i,j)}$.

 i1 : G = graph({{a,b},{b,c},{c,d},{a,d}}) o1 = Graph{a => {b, d}} b => {a, c} c => {b, d} d => {a, c} o1 : Graph i2 : R = gaussianRing G o2 = R o2 : PolynomialRing i3 : compactMatrixForm =false; i4 : K = undirectedEdgesMatrix(R) o4 = | k k 0 k | | a,a a,b a,d | | | | k k k 0 | | a,b b,b b,c | | | | 0 k k k | | b,c c,c c,d | | | | k 0 k k | | a,d c,d d,d | 4 4 o4 : Matrix R <--- R

For mixed graphs with other types of edges, the size of the matrix coincides with the number of elements in compU, 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; i7 : K = undirectedEdgesMatrix(R) o7 = | k k | | 1,1 1,2 | | | | k k | | 1,2 2,2 | 2 2 o7 : Matrix R <--- R