Desktop computer crashing - is my graphics card the issue?

Have you overclocked your gpu?
Have you tried to reinstall your drivers?

I recommend 2 things. First repair restore the operating system, it's probably got corruption issues...
Start the computer in normal windows, disable (turn OFF) ALL the security, firewall...
connect all your hardware.
Put the win 7 disk in the drive, let it start, select "upgrade."
Upgrade will restore all the OS files without erasing your personal files and programs. The OS will be set back to default state.
This will also configure the system for any hardware you have added or subtracted.
This will also fix corrupted files.
After you have restored the OS, do the power profiles (very important).
Click Start, Click Control Panel,
Look at the top of the window, in the path bar you see “control panel >”
Click on “>” (in the path bar) now click on “all control panel options.”
(This will open up all the hidden controls available)
Click Power Options
click on the arrow to “show other plans”
Check the Box that says "high performance"
Click (in high performance) "change plan settings"
Turn off display: set to NEVER
Put the computer to sleep: set to NEVER
Click: Change advanced plan settings
Scroll down the list: Click on the + signs to expand the choices for each item on the list.
Require a password on wake up: set to NO
Hard disk: turn off the hard disk: set to NEVER
Wireless adapter settings:
Sleep: set to NEVER
Allow Hybrid sleep: set to NEVER
Hibernate after: set to NEVER
Allow wake timers: set to disable
USB settings:
USB selective suspend setting: set to NEVER
Power Buttons and lid:
Power button action: Setting: set to shut down
Sleep Button Action: set to: do nothing
PCI Express:
Link State Power Management, Setting: OFF
Processor Power Management: Minimum state (set to) 7%

System Cooling Policy: setting: Active
Maximum State (set to) 100%
Display
Turn off display after: setting: NEVER (turning off display automatically can cause freezing also)
Turn off the monitor power manually, when you want it off. Don’t use the auto monitor turn off.
Multimedia Settings:
When Sharing Media: Setting: Prevent idling to sleep
When Playing Video: Setting: Optimize Video
Click APPLY
Click OK

Open the bios set up and make sure "cool and quiet" is OFF. (AMD)
If there is a power saver or a "quiet mode" in the bios, shut it off...
There may be a performance setting in the bios setup you have...make sure it's cranked up to max.
in the bios, see that the allocation for video, if available, is maxed.

Now open the hardware manager profiles...
click start
click computer
click system properties
click device manager
double click on mice and other pointing devices
right click on HID compliant mouse
left click on properties
click on the power management tab
UN-check the box that says: "allow the computer to turn off this device to save power." (there is now NO check mark in this box)
click OK

Now repeat this procedure for all mice, monitors, keyboards, and ALL USB ports on the device manager list.

You must open ALL the devices one at a time, as above, and turn off the power saver, for each device.

NOW turn all the security back ON. NOW open your security antivirus. Make sure the antivirus is set to "gaming mode." Or "multimedia mode."
This prevents the security updating from interrupting your game / multimedia priority.
IF the security does not have "gaming mode" or "multimedia mode" get different security.
IF you are using "free" security downloaded from the internet, get rid of it NOW.
Use ONLY professional all in one security. DO NOT load multiple mismatched security programs, which conflict with each other.
DO NOT load free tools into your system such as: "driver sweeper" or any of that "free" goofy stuff.

clean the heat sinks and fans. remove the heat sinks from the graphics card and the processor ( video first ) clean up and reapply thermal paste. test.

Make a testing plan. More accurately, make several sub plans. Sounding more like homework, eh? But don't worry these plans will evolve as you go. But WRITE THEM DOWN so 1) they can be repeatable without variance 2) you can analyze them for logic and methodology later, 3) it's doubtful this will be resolved in a single sitting so you have to log.

Pick a few of your favorite barn - burner testing tools to provide repeatable testing. A round of gaming won't cut it you need robo-percision guided torture testing at your disposal. They need to cover the spectrum of possible hardware culprits as well. While it is certainly GPU related at the very least in the sense that when your system takes a dump, your display does that. There's no way to be sure at this point that it doesn't stem from excessive heat or voltage somewhere on the MOB connections i.e. PCI, CPU, Southbridge, DRAM or the graphics card itself.

I jumped in my chair after recognizing the picture you posted. I had to do one of those irrational double takes to make sure it wasn't for my own system (and you were breaking into my home to shoot pictures of it lol ). I have a 5830 GPU and a Samsung PX2370 display like yours. In fact my visit to Tom's now is due to the exact same FAIL image about 10 minutes ago. I'm overclocking my Phenom 1055t for the first time manually.

Definitely clean it back to stock condition first thing. as suggested. I ended up doing a complete tear-down and rebuild because my original install work was pretty lame. Especiallly in the wiring, it was a hornet's nest. I was able to rethink the big picture while putting it back together and ended up with a number of small but cumulative improvements in air flow patterns and resistance. Relying on air cooling, I'm dreading the Summer and it's pesky increased temperatures. While your case is open, see if there are quick and easy changes you could make to wire and excess connector routing. See if you can reorder your drives to provide the clearest path for air flow. Check fans for placement and proper installation: solid connects at both ends, facing the right direction (yes, it happens, i should know).

Good luck. Keep us posted with results of your research, Professor