Recently i've removed the expired McAfee software from my PC and downloaded the trial version of ZoneAlarm. I cannot remove the McAfee security center software or the shortcut. And thus I even cannot run Zone Alarm.
Please help me out.

I assume you tried ADD/REMOVE programs in control panel? Did you check the working directory (Program Files)... many programs leave remnants like personal settings, shortcuts etc. which usually can safely be deleted afterwards.

You can also run in safe mode or real mode dos to remove pesty software. Otherwise, some people use UNinstaller programs or do registry sweep to remove orphan entries. Just have to be careful if do the latter.