let $r$ be a commutative ring with zero-divisor set $z(r)$. the total graph of $r$, denoted by$t(gamma(r))$, is the simple (undirected) graph with vertex set  $r$ where two distinct vertices areadjacent if their sum lies in $z(r)$.this work considers minimum zero-sum $k$-flows for $t(gamma(r))$.both for $vert rvert$ even and the case when $vert rvert$ is odd and $z(g)$ is an ideal of $r$it is s...

recently, e. m'{a}v{c}ajov'{a} and m. v{s}koviera proved that every bidirected eulerian graph which admits a nowhere zero flow, admits a nowhere zero $4$-flow. this result shows the validity of bouchet's nowhere zero conjecture for eulerian bidirected graphs. in this paper we prove the same theorem in a different terminology and with a short and simple proof. more precisely, we p...

Given an instance of the minimum cost flow problem, a version of the corresponding inverse problem, called the capacity inverse problem, is to modify the upper and lower bounds on arc flows as little as possible so that a given feasible flow becomes optimal to the modified minimum cost flow problem. The modifications can be measured by different distances. In this article, we consider the capac...

We study the maximum flow problem in directed H-minorfree graphs where H can be drawn in the plane with one crossing. If a structural decomposition of the graph as a clique-sum of planar graphs and graphs of constant complexity is given, we show that a maximum flow can be computed in O(n logn) time. In particular, maximum flows in directed K3,3-minor-free graphs and directed K5-minor-free graph...

For an undirected graph G, a zero-sum flow is an assignment of non-zero real numbers to the edges, such that the sum of the values of all edges incident with each vertex is zero. It has been conjectured that if a graph G has a zero-sum flow, then it has a zero-sum 6-flow. We prove this conjecture and Bouchet’s Conjecture for bidirected graphs are equivalent. Among other results it is shown that...

As an analogous concept of a nowhere-zero flow for directed graphs, we consider zero-sum flows for undirected graphs in this article. For an undirected graph G, 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 values of edges are less than k. We define the zer...

Let D be a t-(v, k, λ) design and let Ni(D), for 1 ≤ i ≤ t, be the higher incidence matrix of D, a (0, 1)-matrix of size ( v i ) × b, where b is the number of blocks of D. A zero-sum flow of D is a nowhere-zero real vector in the null space of N1(D). A zero-sum k-flow of D is a zero-sum flow with values in {±1, . . . ,±(k− 1)}. In this paper we show that every non-symmetric design admits an int...

Let G be a graph. Assume that l and k are two natural numbers. An l-sum flow on a graph G is an assignment of non-zero real numbers to the edges of G such that for every vertex v of G the sum of values of all edges incident with v equals l. An l-sum k-flow is an l-sum flow with values from the set {±1, . . . ,±(k − 1)}. Recently, it was proved that for every r, r > 3, r 6= 5, every r-regular gr...

In this paper, we propose a new method to estimate the FOE (Focus of Expansion) from moving camera image sequence for vehicle motion understanding, which does not rely on optical flow vectors. Motion understanding of the camera is very important in many applications. Most of the conventional methods for detection and estimation of the position of FOE rely on estimation of optical flow vector in...

Let G = (V,E) be a simple undirected graph. For a given set L ⊂ R, a function ω : E −→ L is called an L-flow. Given a vector γ ∈ R , we say that ω is a γ-L-flow if for each v ∈ V , the sum of the values on the edges incident to v is γ(v). If γ(v) = c, for all v ∈ V , then the γ-L-flow is called a c-sum L-flow. In this paper we study the existence of γ-L-flows for various choices of sets L of re...