Characterization of 2-Path Signed Network

Abstract

A signed network is a network where each edge receives a sign: positive or negative. In this paper, we report our investigation on 2-path signed network of a given signed network Σ , which is defined as the signed network whose vertex set is that of Σ and two vertices in Σ 2 are adjacent if there exist a path of length two between them in Σ . An edge ab in Σ 2 receives a negative sign if all the paths of length two between them are negative, otherwise it receives a positive sign. A signed network is said to be if clusterable its vertex set can be partitioned into pairwise disjoint subsets, called clusters, such that every negative edge joins vertices in different clusters and every positive edge joins vertices in the same clusters. A signed network is balanced if it is clusterable with exactly two clusters. A signed network is sign-regular if the number of positive (negative) edges incident to each vertex is the same for all the vertices. We characterize the 2-path signed graphs as balanced, clusterable, and sign-regular along with their respective algorithms. The 2-path network along with these characterizations is used to develop a theoretic model for the study and control of interference of frequency in wireless communication networks.

An algorithmic characterization and spectral analysis of canonical splitting signed graph ξ(Σ)

“…In recent years, researchers have explored the spectral properties of graphs constructed through graph operations such as disjoint union, Cartesian product, Kronecker product, strong product, lexicographic product, corona, edge corona, and neighbourhood corona. A comprehensive overview of results on the spectra of these graphs can be found in the literature [ [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] ] and one can find the properties of derived signed graphs in [ [13] , [14] , [15] , [16] , [17] , [18] ]. In [ 19 ] authors presented the idea of the splitting graph for a given graph .…”