Try this Code:
if (dataGridView1.Rows.Count > 1 && dataGridView1.SelectedRows[0].Index != dataGridView1.Rows.Count - 1)
{
delcmd.CommandText = "DELETE FROM table_Name WHERE Column_Name=" + dataGridView1.SelectedRows[0].Cells[0].Value.ToString() + "";
con.Open();
delcmd.Connection = con;
delcmd.ExecuteNonQuery();
con.Close();
dataGridView1.Rows.RemoveAt(dataGridView1.SelectedRows[0].Index);
}