count the total number of comments the user has received for his published posts

You can use the get_comments() function like so:

$total = get_comments( array(
    'post_author' => get_current_user_id(),
    'post_status' => 'publish',
    'type'        => 'comment',
    'count'       => true,
) );

echo 'total comments of your all posts : ' . $total;

Just customize the arguments based on your requirements/preferences, and in the above example, I’m retrieving the total number of comments for published posts where the author is the currently logged-in user, but only for comments of the comment type.

And if you want, you can use the status argument to limit to certain status only, e.g. approve to count approved comments only.