Zero-Sum Flow Number of Categorical and Strong Product of Graphs

Abstract

A zero-sum flow is an assignment of nonzero integers to the edges such that the sum of the values of all edges incident with each vertex is zero, and we call it a zero-sum $k$-flow if the absolute values of edges are less than $k$. We define the zero-sum flow number of $G$ as the least integer $k$ for which $G$ admitting a zero sum $k$-flow.? In this paper we gave complete zero-sum flow and zero sum numbers for categorical and strong product of two graphs namely cycle and paths.