I have to find the solution that this algorithm gives about graphs, knowing that the Graph is given by $ G = (V, E) $, with arcs labeled by $ w $ and that $ G $ \ $ e $ indicates that the arc $ e $ of the graph. I mean $ G $\ $ e $ is the elgraph with vertices $ V $ and arcs $ E $ \ {e}. Being $ V_G $ for the vertices of $ G $ and $ E_G $ for its arcs, and that the graph is connected.
I understand that the first line orders the values of $ w $ in decreasing order, then it goes through the arcs, but after the if condition I don't understand what the next line does, G $ \leftarrow $ $ G $ \ $ e $.
It goes through the edges in order of decreasing weight. If removing an edge would disconnect the graph, that edge is kept; otherwise, it is deleted. When the algorithm is done, what’s left is a spanning tree for $G$.
To see this, let $G'$ be the graph that is returned by the algorithm. Clearly $G'$ is connected: the algorithm never removes an edge whose removal would disconnect the graph. Suppose that $G'$ contained a cycle $C$. Let $e$ be the first edge of $C$ in the sorted list produced by the first line of the algorithm. When the algorithm reached $e$ in the for loop, all of the edges of $C$ were still in the graph, so removing $e$ would not have disconnected the graph, and therefore $e$ would have been removed. Thus, $G'$ cannot contain a cycle. Being connected and acyclic, $G'$ is a tree and hence a spanning tree for $G$.