Depth and Stanley depth of edge ideals associated to some line graphs

Abstract

In this paper we study depth and Stanley depth of the edge ideals and quotient rings of the edge ideals, associated to classes of graphs obtained by taking the strong product of two graphs. We consider the cases when either both graphs are arbitrary paths or one is an arbitrary path and the other is an arbitrary cycle. We give exact formulae for values of depth and Stanley depth for some subclasses. We also give some sharp upper bounds for depth and Stanley depth in the general cases.

Depth and Stanley depth of the edge ideals of the strong product of some graphs

“…The strange thing about Stanley depth is that it shares some properties and bounds with homological invariant depth see ( [11,15,22,24]). Until now mathematicians are not too much familiar with Stanley depth as it is hard to compute, for computation and some known results we refer the readers to ( [1,12,16,17,19]). Let P n and C n represent path and cycle respectively on n vertices and represents the strong product of two graphs.…”