Stanley Depth of the Edge Ideal of Extended Gear Networks and Application in Circuit Analysis

Abstract

Graph theory is widely used in power network analysis, complex network, and engineering calculation. Stanley depth is a geometric invariant of the module which is closely related to an algebraic invariant called depth of the module. At first, we propose a generalization of classical gear graph and extended $m-$ level gear graph and then establish general closed formulas for the sharp bounds of Stanley depth of quotient of edge ideals associated to extended $m$ -level gear graph. We establish general closed formulas for the sharp bounds of Stanley depth of quotient of edge ideals associated to extended gear graph. We recover these bounds for Stanley depth of the quotient of edge ideals associated to classical gear graphs.