A signed
graph (or sigraph in short) S
is a graph G in which each edge x carries a
value called its sign
denoted specially as . 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. Iterated signed line
graphs
=
k
, S:= is defined similarly. In this paper, we
give an algorithm to obtain iterated line sigraph and detect for which value of
‘k’ it is balanced and
determine its complexity. In the end we will propose a technique that will use
adjacency matrix of S and adjacency matrix of which is balanced for some
‘k’ as a parameter to encrypt a network and
forward the data in the form of balanced and will decrypt it by applying inverse
matrix operations.
A signed graph (or sigraph in short) S is a graph G in which each edge x carries a value s(x) ∈ {+1, −1} called its sign denoted specially as S = (G, s). Given a sigraph S, a new sigraph C E (S), called the common-edge sigraph of S is that sigraph whose vertex-set is the set of pairs of adjacent edges in S and two vertices of C E (S) are adjacent if the corresponding pairs of adjacent edges of S have exactly one edge in common, and the sign of the edge is the sign of the common edge. If all the edges of the sigraph S carry + sign then S is actually a graph and the corresponding commonedge sigraph is termed as the common-edge graph. In this paper, algorithms are defined to obtain a common-edge sigraph and detect whether it is balanced or not in O(n 3 ) steps which will be optimal in nature.
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