[SPARK-47664][PYTHON][CONNECT][TESTS][FOLLOW-UP] Add more tests

### What changes were proposed in this pull request?
Add more tests

### Why are the changes needed?
for test coverage, to address apache#45788 (comment)

### Does this PR introduce _any_ user-facing change?
no, test only

### How was this patch tested?
new tests

### Was this patch authored or co-authored using generative AI tooling?
no

Closes apache#45809 from zhengruifeng/col_name_val_test.

Authored-by: Ruifeng Zheng <[email protected]>
Signed-off-by: Ruifeng Zheng <[email protected]>

python/pyspark/sql/tests/connect/test_connect_basic.py

@@ -1219,6 +1219,11 @@ def test_parse_col_name(self):
         self.assert_eq(parse_attr_name("`a"), None)
         self.assert_eq(parse_attr_name("a`"), None)
 
+        self.assert_eq(parse_attr_name("`a`.b"), ["a", "b"])
+        self.assert_eq(parse_attr_name("`a`.`b`"), ["a", "b"])
+        self.assert_eq(parse_attr_name("`a```.b"), ["a`", "b"])
+        self.assert_eq(parse_attr_name("`a``.b"), None)
+
         self.assert_eq(parse_attr_name("a.b.c"), ["a", "b", "c"])
         self.assert_eq(parse_attr_name("`a`.`b`.`c`"), ["a", "b", "c"])
         self.assert_eq(parse_attr_name("a.`b`.c"), ["a", "b", "c"])
@@ -1284,7 +1289,6 @@ def test_verify_col_name(self):
         self.assertTrue(verify_col_name("m.`s`.id", cdf.schema))
         self.assertTrue(verify_col_name("`m`.`s`.`id`", cdf.schema))
         self.assertFalse(verify_col_name("m.`s.id`", cdf.schema))
-        self.assertFalse(verify_col_name("m.`s.id`", cdf.schema))
 
         self.assertTrue(verify_col_name("a", cdf.schema))
         self.assertTrue(verify_col_name("`a`", cdf.schema))
@@ -1294,6 +1298,47 @@ def test_verify_col_name(self):
         self.assertTrue(verify_col_name("`a`.`v`", cdf.schema))
         self.assertFalse(verify_col_name("`a`.`x`", cdf.schema))
 
+        cdf = (
+            self.connect.range(10)
+            .withColumn("v", CF.lit(123))
+            .withColumn("s.s", CF.struct("id", "v"))
+            .withColumn("m`", CF.struct("`s.s`", "v"))
+        )
+
+        # root
+        # |-- id: long (nullable = false)
+        # |-- v: string (nullable = false)
+        # |-- s.s: struct (nullable = false)
+        # |    |-- id: long (nullable = false)
+        # |    |-- v: string (nullable = false)
+        # |-- m`: struct (nullable = false)
+        # |    |-- s.s: struct (nullable = false)
+        # |    |    |-- id: long (nullable = false)
+        # |    |    |-- v: string (nullable = false)
+        # |    |-- v: string (nullable = false)
+
+        self.assertFalse(verify_col_name("s", cdf.schema))
+        self.assertFalse(verify_col_name("`s`", cdf.schema))
+        self.assertFalse(verify_col_name("s.s", cdf.schema))
+        self.assertFalse(verify_col_name("s.`s`", cdf.schema))
+        self.assertFalse(verify_col_name("`s`.s", cdf.schema))
+        self.assertTrue(verify_col_name("`s.s`", cdf.schema))
+
+        self.assertFalse(verify_col_name("m", cdf.schema))
+        self.assertFalse(verify_col_name("`m`", cdf.schema))
+        self.assertTrue(verify_col_name("`m```", cdf.schema))
+
+        self.assertFalse(verify_col_name("`m```.s", cdf.schema))
+        self.assertFalse(verify_col_name("`m```.`s`", cdf.schema))
+        self.assertFalse(verify_col_name("`m```.s.s", cdf.schema))
+        self.assertFalse(verify_col_name("`m```.s.`s`", cdf.schema))
+        self.assertTrue(verify_col_name("`m```.`s.s`", cdf.schema))
+
+        self.assertFalse(verify_col_name("`m```.s.s.v", cdf.schema))
+        self.assertFalse(verify_col_name("`m```.s.`s`.v", cdf.schema))
+        self.assertTrue(verify_col_name("`m```.`s.s`.v", cdf.schema))
+        self.assertTrue(verify_col_name("`m```.`s.s`.`v`", cdf.schema))
+
 
 if __name__ == "__main__":
     from pyspark.sql.tests.connect.test_connect_basic import *  # noqa: F401