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I purchased a domain name a couple of days ago through yola and was wondering roughly, how long it'll be before the search engines list my site with just the domain and not have yolasite attached afterward?
Congratulations on your new domain! It will take some time for Google to re-spider your site and see that there have been changes. When these changes are detected Google will remove the old domain name from the index and put in the new one. This can take some time, but it will eventually happen and is out of our control.
To expedite the process in the meantime you should re-verify your site with Google Webmaster, resubmit a sitemap, and get new Google Analytics code if you are using Google Analytics. This will allow Google Webmaster and Analytics to begin returning statistics on your new domain.
What you need to do is remove the original Google Webmaster meta tag and the original Google Analytics code that you put into your pages. The tutorials linked above will walk you through the process again.