set_intersection failed, can't see why

I'm trying to build a class of a graph, which is represented by edges and vertices.

I have implemented the edges as a string set, and the vertices as a string set, and would like to create methods in order to create intersections and unions between my created graphs. This is the class:

class Graph {
public:
    string name;
    std::set<string> vertices;
    std::set<string> edges; //<v1,v2> format
    Graph()= default;
    Graph(const Graph&) = default;
    ~Graph() = default;
    Graph graphUnion(const Graph& graph);
    Graph graphIntersection(const Graph& graph);
};

I used the std::set_intersection function for my method:

Graph Graph::graphIntersection(const Graph& graph){
    Graph new_graph;
    std::set<string> new_vertices;
    new_vertices.end() = std::set_intersection(this->vertices.begin(), this->vertices.end(), graph.vertices.begin(),
            graph.vertices.end(), new_vertices.begin());
    std::set<string> new_edges;
    new_edges.end() = std::set_intersection(this->edges.begin(), this->edges.end(), graph.edges.begin(),
                                            graph.edges.end(), new_edges.begin());

    new_graph.vertices = new_vertices;
    new_graph.edges = new_edges;

    return new_graph;
}

Yet when I try to run it I get the following error, which has something to do with the set_intersection function in

/Library/Developer/CommandLineTools/usr/bin/../include/c++/v1/algorithm:5415:27: error: no viable overloaded '='
                *__result = *__first1;
                ~~~~~~~~~ ^ ~~~~~~~~~

Please help me identify this error since I can't tell what is the problem here...

Thank you!