diff --git a/arc/molecule/graph.pyx b/arc/molecule/graph.pyx
index d6382048c9..4440685df2 100644
--- a/arc/molecule/graph.pyx
+++ b/arc/molecule/graph.pyx
@@ -218,18 +218,25 @@ cdef class Graph(object):
 
     cpdef list get_all_edges(self):
         """
-        Returns a list of all edges in the graph.
+        Returns a list of all edges in the graph, each edge appearing once,
+        ordered by the graph's vertex order and, within a vertex, by the order
+        in which its edges were added. The order does not depend on the hash
+        values of the edges, so it is identical in every process.
         """
-        cdef set edge_set
+        cdef list edges
+        cdef set seen
         cdef Vertex vertex
         cdef Edge edge
 
-        edge_set = set()
+        edges = []
+        seen = set()
         for vertex in self.vertices:
             for edge in vertex.edges.values():
-                edge_set.add(edge)
+                if id(edge) not in seen:
+                    seen.add(id(edge))
+                    edges.append(edge)
 
-        return list(edge_set)
+        return edges
 
     cpdef dict get_edges(self, Vertex vertex):
         """
diff --git a/arc/molecule/graph_test.py b/arc/molecule/graph_test.py
index 6a8f11b2c1..fae67189cc 100644
--- a/arc/molecule/graph_test.py
+++ b/arc/molecule/graph_test.py
@@ -1,8 +1,12 @@
 #!/usr/bin/env python3
 # encoding: utf-8
 
+import os
+import subprocess
+import sys
 import unittest
 
+import arc
 from arc.molecule.graph import Edge, Graph, Vertex
 
 
@@ -108,6 +112,37 @@ def test_get_all_edges(self):
         self.assertIsInstance(edges, list)
         self.assertEqual(len(edges), 5)
 
+    def test_get_all_edges_orders_the_edges_by_vertex(self):
+        """
+        Test that Graph.get_all_edges() returns the edges in vertex order, each edge once.
+        """
+        expected = []
+        for vertex in self.graph.vertices:
+            for edge in vertex.edges.values():
+                if not any(edge is seen for seen in expected):
+                    expected.append(edge)
+        self.assertEqual(self.graph.get_all_edges(), expected)
+
+    def test_get_all_edges_order_does_not_depend_on_the_hash_seed(self):
+        """
+        Test that Graph.get_all_edges() returns the same order in processes with different hash seeds.
+        """
+        script = ('from arc.molecule.molecule import Molecule\n'
+                  'mol = Molecule(smiles="c1ccccc1Cc1ccccc1")\n'
+                  'atoms = mol.atoms\n'
+                  'print([(atoms.index(edge.vertex1), atoms.index(edge.vertex2)) '
+                  'for edge in mol.get_all_edges()])\n')
+        orders = list()
+        for seed in ('1', '35'):
+            arc_root = os.path.dirname(os.path.dirname(os.path.abspath(arc.__file__)))
+            environment = dict(os.environ, PYTHONHASHSEED=seed, PYTHONPATH=arc_root)
+            result = subprocess.run([sys.executable, '-c', script], capture_output=True, text=True,
+                                    env=environment, timeout=300)
+            self.assertEqual(result.returncode, 0, f'Subprocess failed: {result.stderr}')
+            orders.append(result.stdout.strip())
+        self.assertTrue(orders[0])
+        self.assertEqual(orders[0], orders[1])
+
     def test_has_vertex(self):
         """
         Test the Graph.has_vertex() method.