diff --git a/lucene/core/src/java/org/apache/lucene/util/hnsw/HnswGraphBuilder.java b/lucene/core/src/java/org/apache/lucene/util/hnsw/HnswGraphBuilder.java
index b308a86fa91..d1161798117 100644
--- a/lucene/core/src/java/org/apache/lucene/util/hnsw/HnswGraphBuilder.java
+++ b/lucene/core/src/java/org/apache/lucene/util/hnsw/HnswGraphBuilder.java
@@ -104,6 +104,12 @@ public final class HnswGraphBuilder {
     if (searchStrategy == VectorValues.SearchStrategy.NONE) {
       throw new IllegalStateException("No distance function");
     }
+    if (maxConn <= 0) {
+      throw new IllegalArgumentException("maxConn must be positive");
+    }
+    if (beamWidth <= 0) {
+      throw new IllegalArgumentException("beamWidth must be positive");
+    }
     this.maxConn = maxConn;
     this.beamWidth = beamWidth;
     boundedVectors = new BoundedVectorValues(vectorValues);
diff --git a/lucene/core/src/test/org/apache/lucene/index/TestKnnGraph.java b/lucene/core/src/test/org/apache/lucene/index/TestKnnGraph.java
index d7f8fc91248..034b4e3f0d3 100644
--- a/lucene/core/src/test/org/apache/lucene/index/TestKnnGraph.java
+++ b/lucene/core/src/test/org/apache/lucene/index/TestKnnGraph.java
@@ -29,6 +29,7 @@ import org.apache.lucene.store.Directory;
 import org.apache.lucene.util.LuceneTestCase;
 
 import org.apache.lucene.util.hnsw.HnswGraphBuilder;
+import org.junit.After;
 import org.junit.Before;
 
 import java.io.IOException;
@@ -47,9 +48,20 @@ public class TestKnnGraph extends LuceneTestCase {
 
   private static final String KNN_GRAPH_FIELD = "vector";
 
+  private static int maxConn;
+
   @Before
   public void setup() {
     randSeed = random().nextLong();
+    if (random().nextBoolean()) {
+      maxConn = HnswGraphBuilder.DEFAULT_MAX_CONN;
+      HnswGraphBuilder.DEFAULT_MAX_CONN = random().nextInt(1000) + 1;
+    }
+  }
+
+  @After
+  public void cleanup() {
+    HnswGraphBuilder.DEFAULT_MAX_CONN = maxConn;
   }
 
   /**
@@ -114,6 +126,18 @@ public class TestKnnGraph extends LuceneTestCase {
     }
   }
 
+  private void dumpGraph(KnnGraphValues values, int size) throws IOException {
+    for (int node = 0; node < size; node++) {
+      int n;
+      System.out.print("" + node + ":");
+      values.seek(node);
+      while ((n = values.nextNeighbor()) != NO_MORE_DOCS) {
+        System.out.print(" " + n);
+      }
+      System.out.println();
+    }
+  }
+
   // TODO: testSorted
   // TODO: testDeletions
 
@@ -223,7 +247,6 @@ public class TestKnnGraph extends LuceneTestCase {
         int[][] graph = new int[reader.maxDoc()][];
         boolean foundOrphan= false;
         int graphSize = 0;
-        int node = -1;
         for (int i = 0; i < reader.maxDoc(); i++) {
           int nextDocWithVectors = vectorValues.advance(i);
           //System.out.println("advanced to " + nextDocWithVectors);
@@ -236,7 +259,7 @@ public class TestKnnGraph extends LuceneTestCase {
             break;
           }
           int id = Integer.parseInt(reader.document(i).get("id"));
-          graphValues.seek(++node);
+          graphValues.seek(graphSize);
           // documents with KnnGraphValues have the expected vectors
           float[] scratch = vectorValues.vectorValue();
           assertArrayEquals("vector did not match for doc " + i + ", id=" + id + ": " + Arrays.toString(scratch),
@@ -267,10 +290,14 @@ public class TestKnnGraph extends LuceneTestCase {
         } else {
           assertTrue("Graph has " + graphSize + " nodes, but one of them has no neighbors", graphSize > 1);
         }
-        // assert that the graph in each leaf is connected and undirected (ie links are reciprocated)
-        // assertReciprocal(graph);
-        assertConnected(graph);
-        assertMaxConn(graph, HnswGraphBuilder.DEFAULT_MAX_CONN);
+        if (HnswGraphBuilder.DEFAULT_MAX_CONN > graphSize) {
+          // assert that the graph in each leaf is connected and undirected (ie links are reciprocated)
+          assertReciprocal(graph);
+          assertConnected(graph);
+        } else {
+          // assert that max-connections was respected
+          assertMaxConn(graph, HnswGraphBuilder.DEFAULT_MAX_CONN);
+        }
         totalGraphDocs += graphSize;
       }
     }
@@ -333,6 +360,9 @@ public class TestKnnGraph extends LuceneTestCase {
         }
       }
     }
+    for (int i = 0; i < count; i++) {
+      assertTrue("Attempted to walk entire graph but never visited " + i, visited.contains(i));
+    }
     // we visited each node exactly once
     assertEquals("Attempted to walk entire graph but only visited " + visited.size(), count, visited.size());
   }
diff --git a/lucene/core/src/test/org/apache/lucene/util/hnsw/TestHnsw.java b/lucene/core/src/test/org/apache/lucene/util/hnsw/TestHnsw.java
index 5a1b7325b7b..8f50a1dac59 100644
--- a/lucene/core/src/test/org/apache/lucene/util/hnsw/TestHnsw.java
+++ b/lucene/core/src/test/org/apache/lucene/util/hnsw/TestHnsw.java
@@ -456,4 +456,10 @@ public class TestHnsw extends LuceneTestCase {
         assertTrue(min.check(f + 1e-5f)); // delta is zero initially
     }
 
+    public void testHnswGraphBuilderInvalid() {
+        expectThrows(NullPointerException.class, () -> new HnswGraphBuilder(null, 0, 0, 0));
+        expectThrows(IllegalArgumentException.class, () -> new HnswGraphBuilder(new RandomVectorValues(1, 1, random()), 0, 10, 0));
+        expectThrows(IllegalArgumentException.class, () -> new HnswGraphBuilder(new RandomVectorValues(1, 1, random()), 10, 0, 0));
+    }
+
 }