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###### (β ,α)−Connectivity Index of Graphs

1 Introduction Let G = ( V,E ) be a simple graph with n = | V | vertices and m = | E |edges. As usual, n is said to be an order and m the size of G . The subdivision graph S ( G ) is the graph obtained from G by replacing each edge by a path of length 2. The line graph L ( G ) of G is the graph whose vertex set is E ( G ) in which two vertices are adjacent if and only if they are adjacent in G . The tadpole graph T n,k is the graph obtained by joining a cycle of n vertices with a path of length k . The cartesian product G × H of graphs

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###### Designing optimal trajectories for a skimmer ship to clean, recover and prevent the oil spilled on the sea from reaching the coast

and analyse the behaviour of the solution regarding some key parameters (as the pumping power of the skimmer). Finally, some conclusions are made in Section 4 2 Materials and Methods 2.1 Eulerian Mathematical Model Mathematical modelling of the transport and diffusion of an oil spill in the sea is of high interest to remediate the environmental impact (e.g., [ 24 , 25 , 26 ]). We have developed an Eulerian model for the case where the density of the pollutant is smaller than that of one of the sea water (so that it remains at the surface) and

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