A circular zero-sum flow for a graph G is a function \(f:E(G) \rightarrow {\mathbb {R}}{\setminus }\{0\}\) such that for every vertex v, \(\sum _{e\in E_v}f(e)=0\), where \(E_v\) is the set of all edges incident with v. If for each edge e, \(1\le |f(e)| \le r-1\), where \(r\ge 2\) is a real number, then f is called a circular zero-sum r-flow. Also, if r is a positive integer and for each edge e, f(e) is an integer, then f is called a zero-sum r-flow. If G has a circular zero-sum flow, then the minimum \(r\ge 2\) for which G has a circular zero-sum r-flow is called the circular zero-sum flow number of G and is denoted by \(\Phi _c(G)\). Also, the minimum integer \(r\ge 2\) for which G has a zero-sum r-flow is called the flow number for G and is denoted by \(\Phi (G)\). In this paper, we investigate circular zero-sum r-flows of regular graphs. In particular, we show that if G is k-regular with m edges, then \(\Phi _c(G)=2\) for even k and even m, \(\Phi _c(G)=1+\frac{k+2}{k-2}\) for even k and odd m, and \(\Phi _c(G)\le 1+(\frac{k+1}{k-1})^2\) for odd k. It was proved that for every k-regular graph G with \(k\ge 3\), \(\Phi (G)\le 5\). Here, using circular zero-sum flows, we present a new proof of this result when \(k \ne 5\). Finally, we prove that a graph G has a circular zero-sum flow f such that for every edge e, \(l(e) \le f(e) \le u(e)\), if and only if for every partition of V(G) into three subsets A, B, C,

where l(A, C) is the sum of values of l on the edges between A, C, and l(A) is the sum of values of l on the edges with both ends in A (the definitions of u(B, C) and u(B) are analogous).

图G的圆零和流是一个函数\（f：E（G）\ rightarrow {\ mathbb {R}} {\ setminus} \ {0 \} \），因此对于每个顶点v，\（ \ sum _ {e \ in E_v} f（e）= 0 \），其中\（E_v \）是与v入射的所有边的集合。如果对于每个边缘ë，\（1 \文件| F（e）中| \文件的r-1 \） ，其中\（R \ GE 2 \）是实数，则˚F称为圆形零和- [R -流。同样，如果r是一个正整数，并且对于每个边e，f（e）是一个整数，则f称为零和r流。如果ģ具有圆形零和流量，则最低\（R \ GE 2 \）为其ģ具有圆形零和- [R -flow被称为的圆形零和流号码ģ并且由表示\（ \ Phi _c（G）\）。此外，最小整数\（R \ GE 2 \）为其ģ具有零和- [R -flow被称为用于流数ģ并且被表示为\（\披（G）\） 。在本文中，我们研究了正则图的圆零和r流。特别地，我们表明如果G为k-具有m个边的规则数，然后\（\ Phi _c（G）= 2 \）对于偶数k和偶数m，\（\ Phi _c（G）= 1 + \ frac {k + 2} {k-2} \ ）表示偶数k和奇数m，而\（\ Phi _c（G）\ le 1 +（\ frac {k + 1} {k-1}）^ 2 \）表示奇数k。据证明，对于每ķ -regular图ģ与\（K \ GE 3 \） ，\（\披（G）\文件5 \） 。在这里，使用循环零和流，当\（k \ ne 5 \）时，我们给出了这个结果的新证明。最后，我们证明图G的循环零和流量为f这样，对于且仅当V（G）分为三个子集A，  B，  C时，对于每个边e，\（l（e）\ le f（e）\ le u（e）\）

其中l（A，  C）是A，  C之间的边缘上l的值的和，l（A）是两端在A中的边缘上的l的值的和（u（B，  C）和u（B）类似。