Handle edge cases in Laplacian centrality (networkx#6938)

* networkx#6571 laplacian_centrality not working for graphs with single nodes

* networkx#6571 PR code review comments.

* Minor test cleanup.

* More minor test cleanup.

* avoid checking == 0.0, move up check and use empty_graph in tests

---------

Co-authored-by: Pieter Kuppens <pkuppens@users.noreply.github.com>
Co-authored-by: Dan Schult <dschult@colgate.edu>

Author: 3 people

networkx/algorithms/centrality/laplacian.py

@@ -80,6 +80,8 @@ def laplacian_centrality(
     ------
     NetworkXPointlessConcept
         If the graph `G` is the null graph.
+    ZeroDivisionError
+        If the graph `G` has no edges (is empty) and normalization is requested.
 
     References
     ----------
@@ -98,8 +100,12 @@ def laplacian_centrality(
 
     if len(G) == 0:
         raise nx.NetworkXPointlessConcept("null graph has no centrality defined")
+    if G.size(weight=weight) == 0:
+        if normalized:
+            raise ZeroDivisionError("graph with no edges has zero full energy")
+        return {n: 0 for n in G}
 
-    if nodelist != None:
+    if nodelist is not None:
         nodeset = set(G.nbunch_iter(nodelist))
         if len(nodeset) != len(nodelist):
             raise nx.NetworkXError("nodelist has duplicate nodes or nodes not in G")
@@ -126,7 +132,11 @@ def laplacian_centrality(
         new_diag = lap_matrix.diagonal() - abs(lap_matrix[:, i])
         np.fill_diagonal(A_2, new_diag[all_but_i])
 
-        new_energy = np.power(sp.linalg.eigh(A_2, eigvals_only=True), 2).sum()
+        if len(all_but_i) > 0:  # catches degenerate case of single node
+            new_energy = np.power(sp.linalg.eigh(A_2, eigvals_only=True), 2).sum()
+        else:
+            new_energy = 0.0
+
         lapl_cent = full_energy - new_energy
         if normalized:
             lapl_cent = lapl_cent / full_energy

networkx/algorithms/centrality/tests/test_laplacian_centrality.py

@@ -6,6 +6,38 @@
 sp = pytest.importorskip("scipy")
 
 
+def test_laplacian_centrality_null_graph():
+    G = nx.Graph()
+    with pytest.raises(nx.NetworkXPointlessConcept):
+        d = nx.laplacian_centrality(G, normalized=False)
+
+
+def test_laplacian_centrality_single_node():
+    """See gh-6571"""
+    G = nx.empty_graph(1)
+    assert nx.laplacian_centrality(G, normalized=False) == {0: 0}
+    with pytest.raises(ZeroDivisionError):
+        nx.laplacian_centrality(G, normalized=True)
+
+
+def test_laplacian_centrality_unconnected_nodes():
+    """laplacian_centrality on a unconnected node graph should return 0
+
+    For graphs without edges, the Laplacian energy is 0 and is unchanged with
+    node removal, so::
+
+        LC(v) = LE(G) - LE(G - v) = 0 - 0 = 0
+    """
+    G = nx.empty_graph(3)
+    assert nx.laplacian_centrality(G, normalized=False) == {0: 0, 1: 0, 2: 0}
+
+
+def test_laplacian_centrality_empty_graph():
+    G = nx.empty_graph(3)
+    with pytest.raises(ZeroDivisionError):
+        d = nx.laplacian_centrality(G, normalized=True)
+
+
 def test_laplacian_centrality_E():
     E = nx.Graph()
     E.add_weighted_edges_from(