Many optimization problems can be reduced to the maximum flow problem in a network. However, the maximum flow problem is equivalent to the problem of the minimum cut, as shown by Fulkerson and Ford (Fulkerson & Ford, 1956). There are several algorithms of the graph’s cut, such as the Ford-Fulkerson algorithm (Ford & Fulkerson, 1962), the Edmonds-Karp algorithm (Edmonds & Karp, 1972) or the Goldberg-Tarjan algorithm (Goldberg & Tarjan, 1988). In this paper, we study the parallel computation of the Edmonds-Karp algorithm which is an optimized version of the Ford-Fulkerson algorithm. Indeed, this algorithm, when executed on large graphs, can be extremely slow, with a complexity of the order of O|V|.|E|2 (where |V| designates the number of vertices and |E| designates the number of the edges of the graph). So why we are studying its implementation on inexpensive parallel platforms such as OpenMp and GP-GPU. Our work also highlights the limits for parallel computing on this algorithm.

In this article, we consider the construction of a SVIR (Susceptible, Vaccinated, Infected, Recovered) stochastic compartmental model of measles. We prove that the deterministic solution is asymptotically the average of the stochastic solution in the case of small population size. The choice of this model takes into account the random fluctuations inherent to the epidemiological characteristics of rural populations of Niger, notably a high prevalence of measles in children under 5, coupled with a very low immunization coverage.

We use submultiplicative companion matrix norms to provide new bounds for roots for a given polynomial ( ). From a n n ×Fiedler companion matrix C, sparse companion matrices and triangular Hessenberg matrices are introduced. Then, we identify a special triangular Hessenberg matrix r L , supposed to provide a good estimation of the roots. By application of Gershgorin's theorems to this special matrix in case of submultiplicative matrix norms, some estimations of bounds for roots are made. The obtained bounds have been compared to known ones from the literature precisely Cauchy's bounds, Montel's bounds and Carmichel-Mason's bounds. According to the starting formel of r L , we see that the more we have coefficients closed to zero with a norm less than 1, the more the Sparse method is useful.

In this paper, we combine Graeffe matrices with the classical numerical method of Dandelin-Graeffe to estimate bounds for the moduli of the zeros of polynomials. Furthermore, we give some examples showing significant gain for the convergence towards the polynomials dominant zeros moduli.

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