import bpy

def set_mode(newmo):
    mode = bpy.context.scene.objects.active.mode
    bpy.ops.object.mode_set(mode=newmo)
    return mode


def select_verts_edge_count(max_connected_edges = 5):
    ctx = bpy.context
    scn = ctx.scene
    objs = scn.objects
    obj = objs.active
    mesh = obj.data
    verts = mesh.vertices
    faces = mesh.polygons
    edges = mesh.edges

    mode = set_mode("OBJECT")
    vdict = {v:[] for v in range(len(verts))}
    vdict2 = {v:0 for v in range(len(verts))}
    for e in edges:
        for v in e.vertices:
            vdict[v].append(e)

    for v in vdict:
        vdict2[v] = len(vdict[v])

    for v in vdict2:
        if vdict2[v] > max_connected_edges:
            verts[v].select = True




    set_mode(mode)

select_verts_edge_count(max_connected_edges = 5)