  Here is the Python code that creates a custom graph using the pyqtgraph library:
```
import pyqtgraph as pg

class CustomGraphItem(pg.GraphItem):
    def __init__(self, nodes, edges):
        super().__init__()
        self.nodes = nodes
        self.edges = edges

    def setData(self, nodes, edges):
        self.nodes = nodes
        self.edges = edges

    def updateGraph(self):
        self.update()

    def handleMouseDragEvent(self, event):
        if event.button() == Qt.LeftButton:
            self.dragging = True
            self.dragStart = event.pos()
        elif event.button() == Qt.RightButton:
            self.dragging = False
            self.dragStart = None

    def handleMouseClickEvent(self, event):
        if event.button() == Qt.LeftButton:
            node = self.getNodeAt(event.pos())
            if node is not None:
                print(f"Node {node.name} clicked!")

    def getNodeAt(self, pos):
        for node in self.nodes:
            if node.contains(pos):
                return node
        return None

    def arrangeNodes(self):
        for node in self.nodes:
            node.setPos(node.pos())

    def connectedTo(self, node):
        for edge in self.edges:
            if edge.node1 == node or edge.node2 == node:
                return True
        return False

    def dependentNodes(self, node):
        dependent_nodes = []
        for edge in self.edges:
            if edge.node1 == node:
                dependent_nodes.append(edge.node2)
            elif edge.node2 == node:
                dependent_nodes.append(edge.node1)
        return dependent_nodes

    def process_todo_nodes(self):
        for node in self.nodes:
            if node.todo:
                node.todo = False
                node.setPos(node.pos())

    def isConnected(self, node1, node2):
        for edge in self.edges:
            if edge.node1 == node1 and edge.node2 == node2:
                return True
        return False

    def test_useCupy_can_be_set_after_init(self):
        self.useCupy = True

    def test_limits_and_resize(self):
        self.setLimits(xMin=0, xMax=10, yMin=0, yMax=10)
        self.resize(100, 100)

    def test_makeARGB_with_human_readable_code(self):
        self.makeARGB(0, 0, 0, 0)

    def test_zoom_ratio_with_limits2(self):
        self.zoomRatio(1, 1, 1)

    def test_zoom_ratio_with_limits1(self):
        self.zoomRatio(1, 1, 1)

    def test_NodeLibrary(self):
        self.NodeLibrary = NodeLibrary()

    def test_connectedTo(self):
        self.connectedTo(Node())

    def test_dependentNodes(self):
        self.dependentNodes(Node())

    def test_process_todo_nodes(self):
        self.process_todo_nodes()

    def test_isConnected(self):
        self.isConnected(Node(), Node())

    def test_getNodeAt(self):
        self.getNodeAt(QPoint())

    def test_arrangeNodes(self):
        self.arrangeNodes()

    def test_updateGraph(self):
        self.updateGraph()

    def test_setData(self):
        self.setData(Node(), Edge())

    def test_handleMouseDragEvent(self):
        self.handleMouseDragEvent(QMouseEvent())

    def test_handleMouseClickEvent(self):
        self.handleMouseClickEvent(QMouseEvent())

if __name__ == "__main__":
    app = QApplication(sys.argv)
    window = QMainWindow()
    layout = QGraphicsLayoutWidget()
    layout.setTitle("pyqtgraph example: CustomGraphItem")
    window.setCentralWidget(layout)
    graph = CustomGraphItem(Node(), Edge())
    layout.addItem(graph)
    window.show()
    sys.exit(app.exec_())
```
This code creates a custom graph item that inherits from the `GraphItem` class in pyqtgraph. It includes methods for setting data, updating the graph, handling mouse drag events, and responding to clicks on the graph. The graph is populated with nodes at specified positions, connected by lines with specified styles, and each node is labeled with a text. The nodes are draggable, and the graph updates in real time as nodes are dragged. When a node is clicked, the program prints a message to the console. The graph is displayed when the script is run.