Check if a graph is Eulerian 
Let $G=((2,3,4,5,6,7),E)$ be a graph such that {$x$,$y$} $\in E$ if and only if the product of $x$ and $y$ is even, decide if G is an Eulerian graph.

My attempt
I tried to plot the graph, this is the result:

So, if my deductions are true, this is not an Eulerian graph because it's connected but all the vertices doesn't have an even degree. For example $deg(2)=5$. Moreover, there is no trace of Eulerian trails.
I cannot figure out if this assumptions are presumably correct.

I assume you meant the product is even based on the picture you have drawn. Your conclusions are correct for the drawn graph. It is connected and there are vertices of odd degree so it is not Eulerian.
