Signed Graph - Examples

Examples

• The complete signed graph on n vertices with loops, denoted by ±K_no, has every possible positive and negative edge including negative loops, but no positive loops. Its edges correspond to the roots of the root system C_n; the column of an edge in the incidence matrix (see below) is the vector representing the root.

• The complete signed graph with half-edges, ±K_n', is ±K_n with a half-edge at every vertex. Its edges correspond to the roots of the root system B_n, half-edges corresponding to the unit basis vectors.

• The complete signed link graph, ±K_n, is the same but without loops. Its edges correspond to the roots of the root system D_n.

• An all-positive signed graph has only positive edges. If the underlying graph is G, the all-positive signing is written +G.

• An all-negative signed graph has only negative edges. It is balanced if and only if it is bipartite because a circle is positive if and only if it has even length. An all-negative graph with underlying graph G is written −G.

• A signed complete graph has as underlying graph G the ordinary complete graph K_n. It may have any signs. Signed complete graphs are equivalent to two-graphs, which are of value in finite group theory. A two-graph can be defined as the class of vertex sets of negative triangles in a signed complete graph.