Group Constant-Sum Spectrum of Nearly Regular Graphs

For an undirected graph G , a zero-sum flow is an assignment of nonzero integer weights to the edges such that each vertex has a zero-sum, namely the sum of all incident edge weights with each vertex is zero. This concept is an undirected analog of nowhere-zero flows for directed graphs. We study a more general one, namely constant-sum A -flows, which gives edge weights using nonzero elements of an additive Abelian group A and requires each vertex to have a constant-sum instead. In particular, we focus on two special cases: A = Z k , the finite cyclic group of integer congruence modulo k , and A = Z , the infinite cyclic group of integers. The constant sum under a constant-sum A -flow is called an index of G for short, and the set of all possible constant sums (indices) of G is called the constant sum spectrum. It is denoted by I k ( G ) and I ( G ) for A = Z k and A = Z , respectively. The zero-sum flows and constant-sum group flows for regular graphs regarding cases Z and Z k have been studied extensively in the literature over the years. In this article, we study the constant sum spectrum of nearly regular graphs such as wheel graphs W n and fan graphs F n in particular. We completely determine the constant-sum spectrum of fan graphs and wheel graphs concerning Z k and Z , respectively. Some open problems will be mentioned in the concluding remarks.