Characterization of Signed Graphs whose Iterated Signed Line Graphs are Balanced or S−Consistent

A signed graph is a graph in which every edge is designated to be either positive or negative; it is balanced if every cycle contains an even number of negative edges. A marked signed graph is a signed graph each vertex of which is designated to be positive or negative, and it is consistent if every cycle in the signed graph possesses an even number of negative vertices. Signed line graph L(S) of a given signed graph S = (G, σ), as given by Behzad and Chartrand [9], is the signed graph with the standard line graph L(G) of G as its underlying graph and whose edges are assigned the signs according to the rule: for any eiej ∈ E(L(S)), eiej ∈ E−L(S)⇔ the edges ei and ej of S are both negative in S. Iterated signed line graphs Lk(S)=L(Lk−1(S)), k ∈ N , S:= L(S) is defined similarly. Further, L(S) is S−consistent if to each vertex e of L(S), which is an edge of S, one assigns the sign σ(e) then the resulting marked signed graph (L(S))μ is consistent. In this paper, we give a characterization of signed graphs S whose iterated signed line graphs L(S) are balanced or S−consistent. AMS SUBJECT CLASSIFICATION: 2000 Mathematics Subject Classification: Primary 05C 22; Secondry 05C 75.