Signed graphs connected with the root lattice

For any base of the root lattice (An) we can construct a signed graph. A signed graph is one whose edges are signed by +1 or -1. A signed graph is balanced if and only if its vertex set can be divided into two sets-either of which may be empty–so that each edge between the sets is negative and each edge within a set is positive. For a given signed graph Tsaranov, Siedel and Cameron constructed the corresponding root lattice. In the present work we have dealt with signed graphs corresponding to the root lattice An. A connected graph is called a Fushimi tree if its all blocks are complete subgraphs. A Fushimi tree is said to be simple when by deleting any cut vertex we have always two connected components. A signed Fushimi tree is called a Fushimi tree with standard sign if it can be transformed into a signed Fushimi tree whose all edges are signed by +1 by switching. Here we have proved that any signed graph corresponding to An is a simple Fushimi tree with standard sign. Our main result is that s simple Fushimi tree with standard sign is contained in the cluster given by a line.