2-Path Signed Graph of Signed Smith Graphs and Its Applications in Nonlinear Dynamics

The 2-path signed graph derived from a signed graph \(\Sigma =(G,\sigma )\) utilizes \(G^2\) as its base graph. In this derived graph, the sign of an edge \(\{u,v\}\) is determined as follows: if every \(u-v\) path of length 2 within \(\Sigma \) carries a negative sign for all its edges, the sign of \(\{u,v\}\) is set to −1; otherwise, it is set to +1. Smith graphs are defined as graphs having the highest eigenvalue of 2. There are six connected classes of such graphs available in the literature. The study of dynamical systems using signed graphs is a thrust area in signed graph theory and has been used to model biological networks, epidemics, social networks, etc. The outcome of the present work finds a natural application by contributing significantly to the study of Graph Dynamical systems. Derived graphs are useful tools for studying changes in an existing system. 2-path graphs form an important class of derived graphs. This article explores the canonical consistency and equilibrium of 2-path graphs derived from signed Smith graphs.