An Algorithm to detect balancing of iterated line sigraph

Area of Science:

• Graph Theory
• Discrete Mathematics
• Network Security

Background:

• Introduces signed graphs (sigraphs) where edges have assigned values (signs).
• Defines the line sigraph (L(S)) as a sigraph where edges of the root sigraph become vertices.
• Explains iterated signed line graphs (S_k) as a recursive construction of sigraphs.

Purpose of the Study:

• To develop an algorithm for generating iterated line sigraphs.
• To identify the conditions and values of 'k' for which iterated line sigraphs are balanced.
• To determine the computational complexity of these operations.

Main Methods:

• Development of a novel algorithm to compute iterated line sigraphs.
• Analysis of graph properties to detect 'balanced' sigraphs for specific 'k' values.
• Investigation of computational complexity associated with the algorithm.

Main Results:

• An algorithm is presented for constructing iterated line sigraphs.
• Criteria for determining when an iterated line sigraph becomes balanced are established.
• The complexity of the proposed algorithm is analyzed.

Conclusions:

• The study provides a method for generating and analyzing iterated line sigraphs.
• A novel network encryption and decryption technique is proposed using balanced sigraphs and adjacency matrices.
• This research contributes to graph theory and offers potential applications in secure data transmission.