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 |
The object undirectedEdgesMatrix is a method function.