A Characterization of Line Sigraphs

A signed graph (or in short, sigraph) is an ordered pair S = (Su, s) where Su is a graph G = (V,E) called the underlying graph of S and s : E(Su) → {+, -} is a function denned on the edge set E(Su) = E into set {+, −}, called a signing of G We let E+(S) = [e ∈ E(G): s(e) = +} and E−(S) = [e ∈ E(G) : s(e) = -}. Then the set E(S) = E+(S) U E−(S) is called the edge set of S, the elements of E+(S)(E−(S)) are called positive (negative) edges in S. In this way a graph may be regarded as a sigraph in which all the edges are positive; hence we regard graphs as all-positive sigraph (all-negative sigraphs are denned similarly). A sigraph is said to be homogeneous if it is either all-positive or all-negative and heterogenous otherwise. sigraph S, its line sigraph whose vertex set V(L(S)) is the edge set E(S) = E(Su) of S and two vertices of L(S) are joined by a negative edge if and only if they correspond to adjacent negative edges in S. s paper, we define a given sigraph S to be a line sigraph if there exists a sigraph H such that L(H) ≅ S(read as L(H) is isomorphic to S). We then give the following structural characterization of line sigraphs, extending the well known characterization of line graphs. m: A signed graph S is a line sigraph if and only if is a line graph and (ii) If uv is a positive edge of S then either there is no negative edge incident at u or there is no negative edge incident at v.