diff --git a/bellman_ford/DEMO.ipynb b/bellman_ford/DEMO.ipynb
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+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "7f235e2e",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import networkx as nx\n",
+    "from scipy.sparse.csgraph import bellman_ford"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "7d256792",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Shortest distances from vertex A (0): [0, 2, 1, 3, 4]\n"
+     ]
+    }
+   ],
+   "source": [
+    "# edge (u, v, w) means an edge from u to v with weight w\n",
+    "edges = [\n",
+    "    (0, 1, 3),\n",
+    "    (0, 2, 1),\n",
+    "    (1, 2, 7),\n",
+    "    (1, 3, 5),\n",
+    "    (2, 3, 2),\n",
+    "    (3, 2, -1),\n",
+    "    (3, 4, 1),\n",
+    "    (4, 1, -2),\n",
+    "]\n",
+    "\n",
+    "# assuming A-E are indexed as 0-4\n",
+    "distances = [float('inf')] * 5\n",
+    "distances[0] = 0\n",
+    "\n",
+    "for _ in range(4):   # v-1 iterations\n",
+    "    for u, v, w in edges:   # 'w' is the 'new' weight\n",
+    "        if distances[u] < float('inf'):\n",
+    "            distances[v] = min(distances[v], distances[u] + w)\n",
+    "\n",
+    "print(\"Shortest distances from vertex A (0):\", distances)\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "9afe182f",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "nodes: ['A', 'B', 'C', 'D', 'E']\n",
+      "distances from source (A): [0. 2. 1. 3. 4.]\n",
+      "predecessors: [-9999     4     0     2     3]\n",
+      "  A: distance=0.0, predecessor=None\n",
+      "  B: distance=2.0, predecessor=E\n",
+      "  C: distance=1.0, predecessor=A\n",
+      "  D: distance=3.0, predecessor=C\n",
+      "  E: distance=4.0, predecessor=D\n"
+     ]
+    }
+   ],
+   "source": [
+    "# build a small directed graph with weights (same as in example)\n",
+    "G = nx.DiGraph()\n",
+    "G.add_weighted_edges_from([\n",
+    "    ('A', 'B', 3),\n",
+    "    ('A', 'C', 1),\n",
+    "    ('B', 'C', 7),\n",
+    "    ('B', 'D', 5),\n",
+    "    ('C', 'D', 2),\n",
+    "    ('D', 'C', -1),\n",
+    "    ('D', 'E', 1),\n",
+    "    ('E', 'B', -2),\n",
+    "])\n",
+    "\n",
+    "# convert to an adjacency/weight matrix\n",
+    "W = nx.to_numpy_array(G, weight='weight')\n",
+    "\n",
+    "# run scipy's bellman_ford on the weight matrix from source index 0\n",
+    "distances, predecessors = bellman_ford(W, indices=0, return_predecessors=True)\n",
+    "\n",
+    "nodes = list(G.nodes())\n",
+    "print(\"nodes:\", nodes)\n",
+    "print(\"distances from source (A):\", distances)\n",
+    "print(\"predecessors:\", predecessors)\n",
+    "\n",
+    "# interpret results in terms of node names\n",
+    "for i, d in enumerate(distances):\n",
+    "    pred = predecessors[i]\n",
+    "    pred_name = nodes[pred] if pred != -9999 else None\n",
+    "    print(f\"  {nodes[i]}: distance={d}, predecessor={pred_name}\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "358750df",
+   "metadata": {},
+   "source": [
+    "### References\n",
+    "* [algorithms for cp](https://cp-algorithms.com/graph/bellman_ford.html)"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "venv (3.11.0)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.11.0"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}