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Given a sigraph S, H = L(S) called the line sigraph of S is that sigraph in which edges of S are represented as vertices, two of these vertices are defined to be adjacent whenever the corresponding edges in S have a vertex in common and any such edge ef is defined to be negative whenever both e and f are negative edges in S. Here S is called root sigraph of H. For a given sigraph H, H is sigraph marked if the vertices of H are marked with the signs of the distinct edges of some sigraph M, called a marker of H, in a one-to-one manner such that adjacent vertices of H receive the signs of adjacent edges of M; (often, it is convenient to say that H is M-Marked when M is to be specified); H is then said to be S-Consistent whenever a marker S is to be specified, if there is an S-Marking μ of H with respect to which the marked sigraph Hμ is consistent. In this paper, we give an algorithm to obtain a line sigraph and detect whether it is S-Consistent or not and determine its complexity. In the end we will propose a technique that will use adjacency matrix of S and adjacency matrix of L(S) which is S-Consistent as a parameter to encrypt an image and forward the data in the form of S-Consistent L(S) and will decrypt it by applying inverse image and matrix operations. [ABSTRACT FROM AUTHOR]

A signed graph (or sigraph in short) is an ordered pair S = (Su, σ), where Su is a graph G = (V,E), called the underlying graph of S and σ : E → {+1, -1} is a function from the edge set E of Su into the set {+1, -1}, called the signature of S. A marking of a given sigraph S is a function μ : V (S) → {+1, -1}. The marking μσ defined by μσ(v) = ΠejεEv σ(ej), v ε V (S), is called the canonical marking (or, C-marking in short) of S, where Ev is the set of edges ej incident at v in S. A sigraph S is sign-compatible if there exists a marking μ of its vertices such that the end vertices of every negative edge receive '-1' marks in μ and no positive edge does so. Any canonically marked sigraph S, is said to be canonically sign-compatible (or, C-sign-compatible), if it is sign-compatible under C-marking. In this paper, we characterize S such that S, its line sigraphs and its ×-line sigraphs are C-sign-compatible. [ABSTRACT FROM AUTHOR]

A signed graph (or sigraph, in short) is an ordered pair S = (Su, σ), where Su is a graph G = (V,E), called the underlying graph of S and σ : E →? {+, -} is a function from the edge set E of Su into the set {+, -}, called the signature of S. A sigraph is also denoted as S = (V,E, σ). The splitting sigraph Γ(S) = (VΓ,EΓ, σ Γ) of a sigraph S = (V,E, σ) is formed as follows: For each vertex v ε V(S), take a new vertex v'. Join v' to all vertices of S adjacent to v and define, σ Γ(μv') = σ(μv), μ ε N(v). The sigraph Γ(S) thus obtained is called the splitting sigraph of S. In this paper, we characterize balanced splitting sigraphs and C-consistent splitting sigraphs. Also, we establish a characterization of splitting sigraphs. [ABSTRACT FROM AUTHOR]