Facebook log in and out of the website?

I am logging in and out of a website with facebook account information. For this, I made my code as follows

For Index.html my code is similar to this

<div id="fb-root"></div>
<script type="text/javascript">
//<![CDATA[
window.fbAsyncInit = function() {
   FB.init({
     appId      : 'XXXXXXXXXXXXXXXX', // App ID
     channelURL : '', // Channel File, not required so leave empty
     status     : true, // check login status
     cookie     : true, // enable cookies to allow the server to access the session
     oauth      : true, // enable OAuth 2.0
     xfbml      : false  // parse XFBML
   });
};
// logs the user in the application and facebook
function login(){
FB.getLoginStatus(function(r){
     if(r.status === 'connected'){
            window.location.href = 'fbconnect.php';
     }else{
        FB.login(function(response) {
                if(response.authResponse) {
              //if (response.perms)
                    window.location.href = 'fbconnect.php';
            } else {
              // user is not logged in
            }
     },{scope:'email'}); // which data to access from user profile
 }
});
}

// Load the SDK Asynchronously
(function() {
   var e = document.createElement('script'); e.async = true;
   e.src = document.location.protocol + '//connect.facebook.net/en_US/all.js';
   document.getElementById('fb-root').appendChild(e);
}());

//]]>
</script>
<a href='#' onclick='login();'>Facebook Login</a>

and in fbconnect.php my code looks like

<?php
    require 'src/facebook.php';
    $facebook = new Facebook(array(
           'appId'  => 'XXXXXXXXXXXXXXXX',
           'secret' => 'XXXXXXXXXXXXXXXXXXXXXXXXXXXX',
           'cookie' => true,
    ));
    $user = $facebook->getUser();
    $loginUrl = $facebook->getLoginUrl();
    //echo "<a href='$loginUrl'>login</a>";

    $logoutUrl = $facebook->getLogoutUrl();
    //echo $loginUrl; 
    if($user){
        $_SESSION['user_info'] = $user; 
        $_SESSION['user_pro']= $facebook->api('/me');
        echo $_SESSION['user_pro']['first_name'];
        echo $_SESSION['user_pro']['last_name'];
        echo '<pre>';
        //print_r($_SESSION);
        echo '</pre>';
    }
    else{
        echo 'not logged in '; 
    }
    echo "<a href='logout.php'>log out </a>"
?>

Just like the logout system, so the user can also log out of the site, I made code for logout.php, like this

<?php
require 'src/facebook.php';  //including the facebook php sdk
$token = $facebook->getAccessToken();
$url = 'https://www.facebook.com/logout.php?next=' . YOUR_SITE_URL .
  '&access_token='.$token;
session_destroy();
header('Location: '.$url);
?>

But its error is shown here, for example

Notice: Undefined variable: facebook in logout.php on line 3 Fatal error: Call to a member function getAccessToken() on a non-object in logout.php on line 3.

Here he easily enters a login, but when he logs out he shows an error. So can someone kindly tell me how to solve this problem? I am really new to facebook app. Therefore, any help and suggestions will be really noticeable. Thanks