diff --git a/pyproject.toml b/pyproject.toml
index 8b86431b..70e15ba0 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -76,6 +76,7 @@ dependencies = [
    "xarray>=2024.10.0",
    "imagecodecs>=2025.8.2,<2026",
    "zarr>=2.6.1",
+   "fast-array-utils",
 ]
 
 [project.optional-dependencies]
diff --git a/src/squidpy/gr/_build.py b/src/squidpy/gr/_build.py
index fedae0c9..e70b356b 100644
--- a/src/squidpy/gr/_build.py
+++ b/src/squidpy/gr/_build.py
@@ -13,11 +13,13 @@
 import pandas as pd
 from anndata import AnnData
 from anndata.utils import make_index_unique
-from numba import njit
+from fast_array_utils import stats as fau_stats
+from numba import njit, prange
 from scanpy import logging as logg
 from scipy.sparse import (
     SparseEfficiencyWarning,
     block_diag,
+    csr_array,
     csr_matrix,
     isspmatrix_csr,
     spmatrix,
@@ -388,7 +390,7 @@ def _build_connectivity(
     if delaunay:
         tri = Delaunay(coords)
         indptr, indices = tri.vertex_neighbor_vertices
-        Adj = csr_matrix((np.ones_like(indices, dtype=np.float64), indices, indptr), shape=(N, N))
+        Adj = csr_matrix((np.ones_like(indices, dtype=np.float32), indices, indptr), shape=(N, N))
 
         if return_distance:
             # fmt: off
@@ -423,7 +425,7 @@ def _build_connectivity(
             col_indices = np.concatenate(col_indices)
 
         Adj = csr_matrix(
-            (np.ones_like(row_indices, dtype=np.float64), (row_indices, col_indices)),
+            (np.ones_like(row_indices, dtype=np.float32), (row_indices, col_indices)),
             shape=(N, N),
         )
         if return_distance:
@@ -438,29 +440,68 @@ def _build_connectivity(
     return Adj
 
 
-@njit
-def outer(indices: NDArrayA, indptr: NDArrayA, degrees: NDArrayA) -> NDArrayA:
-    res = np.empty_like(indices, dtype=np.float64)
-    start = 0
-    for i in range(len(indptr) - 1):
-        ixs = indices[indptr[i] : indptr[i + 1]]
-        res[start : start + len(ixs)] = degrees[i] * degrees[ixs]
-        start += len(ixs)
+@njit(parallel=True, cache=True)
+def _csr_bilateral_diag_scale_helper(
+    mat: csr_array | csr_matrix,
+    degrees: NDArrayA,
+) -> NDArrayA:
+    """
+    Return an array F aligned with CSR non-zeros such that
+    F[k] = d[i] * data[k] * d[j] for the k-th non-zero (i, j) in CSR order.
+
+    Parameters
+    ----------
+
+    data : array of float
+        CSR `data` (non-zero values).
+    indices : array of int
+        CSR `indices` (column indices).
+    indptr : array of int
+        CSR `indptr` (row pointer).
+    degrees : array of float, shape (n,)
+        Diagonal scaling vector.
+
+    Returns
+    -------
+    array of float
+        Length equals len(data). Entry-wise factors d_i * d_j * data[k]
+    """
+
+    res = np.empty_like(mat.data, dtype=np.float32)
+    for i in prange(len(mat.indptr) - 1):
+        ixs = mat.indices[mat.indptr[i] : mat.indptr[i + 1]]
+        res[mat.indptr[i] : mat.indptr[i + 1]] = degrees[i] * degrees[ixs] * mat.data[mat.indptr[i] : mat.indptr[i + 1]]
 
     return res
 
 
+def symmetric_normalize_csr(adj: spmatrix) -> csr_matrix:
+    """
+    Return D^{-1/2} * A * D^{-1/2}, where D = diag(degrees(A)) and A = adj.
+
+
+    Parameters
+    ----------
+    adj : scipy.sparse.csr_matrix
+
+    Returns
+    -------
+    scipy.sparse.csr_matrix
+    """
+    degrees = np.squeeze(np.array(np.sqrt(1.0 / fau_stats.sum(adj, axis=0))))
+    if adj.shape[0] != len(degrees):
+        raise ValueError("len(degrees) must equal number of rows of adj")
+    res_data = _csr_bilateral_diag_scale_helper(adj, degrees)
+    return csr_matrix((res_data, adj.indices, adj.indptr), shape=adj.shape)
+
+
 def _transform_a_spectral(a: spmatrix) -> spmatrix:
     if not isspmatrix_csr(a):
         a = a.tocsr()
     if not a.nnz:
         return a
 
-    degrees = np.squeeze(np.array(np.sqrt(1.0 / a.sum(axis=0))))
-    a = a.multiply(outer(a.indices, a.indptr, degrees))
-    a.eliminate_zeros()
-
-    return a
+    return symmetric_normalize_csr(a)
 
 
 def _transform_a_cosine(a: spmatrix) -> spmatrix:
diff --git a/tests/graph/test_spatial_neighbors.py b/tests/graph/test_spatial_neighbors.py
index ac4bd6a6..95165353 100644
--- a/tests/graph/test_spatial_neighbors.py
+++ b/tests/graph/test_spatial_neighbors.py
@@ -355,3 +355,72 @@ def test_mask_graph(
                 negative_mask=True,
                 key_added=key_added,
             )
+
+    def test_spatial_neighbors_transform_mathematical_properties(self, non_visium_adata: AnnData):
+        """
+        Test mathematical properties of each transform.
+        """
+        # Test spectral transform properties
+        spatial_neighbors(non_visium_adata, delaunay=True, coord_type=None, transform="spectral")
+        adj_spectral = non_visium_adata.obsp[Key.obsp.spatial_conn()].toarray()
+
+        # Spectral transform should be symmetric
+        np.testing.assert_allclose(adj_spectral, adj_spectral.T, atol=1e-10)
+
+        # Spectral transform should have normalized rows (L2 norm <= 1)
+        row_norms = np.sqrt(np.sum(adj_spectral**2, axis=1))
+        np.testing.assert_array_less(row_norms, 1.0 + 1e-10)
+
+        # Test cosine transform properties
+        spatial_neighbors(non_visium_adata, delaunay=True, coord_type=None, transform="cosine")
+        adj_cosine = non_visium_adata.obsp[Key.obsp.spatial_conn()].toarray()
+
+        # Cosine transform should be symmetric
+        np.testing.assert_allclose(adj_cosine, adj_cosine.T, atol=1e-10)
+
+        # Cosine transform should have values in [-1, 1]
+        np.testing.assert_array_less(-1.0 - 1e-10, adj_cosine)
+        np.testing.assert_array_less(adj_cosine, 1.0 + 1e-10)
+
+        # Diagonal of cosine transform should be 1 (self-similarity)
+        np.testing.assert_allclose(np.diag(adj_cosine), 1.0, atol=1e-10)
+
+    def test_spatial_neighbors_transform_edge_cases(self, non_visium_adata: AnnData):
+        """
+        Test transforms with edge cases (empty graph, single node, etc.).
+        """
+        # Test with a very small dataset
+        small_adata = non_visium_adata[:5].copy()  # Only 5 points
+
+        # Test all transforms with small dataset
+        for transform in [None, "spectral", "cosine"]:
+            spatial_neighbors(small_adata, delaunay=True, coord_type=None, transform=transform)
+            assert Key.obsp.spatial_conn() in small_adata.obsp
+            assert Key.obsp.spatial_dist() in small_adata.obsp
+
+            # Verify transform parameter is saved
+            assert small_adata.uns[Key.uns.spatial_neighs()]["params"]["transform"] == transform
+
+    def test_spatial_neighbors_spectral_transform_properties(self, non_visium_adata: AnnData):
+        """
+        Test that spectral transform preserves nonzero pattern and normalizes rows to sum to 1.
+        """
+        # Apply spatial_neighbors without transform
+        spatial_neighbors(non_visium_adata, delaunay=True, coord_type=None, transform=None)
+        adj_no_transform = non_visium_adata.obsp[Key.obsp.spatial_conn()].copy()
+
+        # Apply spatial_neighbors with spectral transform
+        spatial_neighbors(non_visium_adata, delaunay=True, coord_type=None, transform="spectral")
+        adj_spectral = non_visium_adata.obsp[Key.obsp.spatial_conn()]
+
+        # Check that nonzero patterns are identical
+        np.testing.assert_array_equal(
+            adj_no_transform.nonzero(),
+            adj_spectral.nonzero(),
+            err_msg="Spectral transform should preserve the sparsity pattern",
+        )
+
+        w = np.linalg.eigvals(adj_spectral.toarray())
+        # Eigenvalues should be in range [-1, 1]
+        np.testing.assert_array_less(w, 1.0, err_msg="Eigenvalues should be <= 1")
+        np.testing.assert_array_less(-1.0, w, err_msg="Eigenvalues should be >= -1")