Stand computer hangs with VGA cable s.Television. Laptop hangs on TV with HDMI connection. Do I still need to connect the PC and the laptop with a LAN cable?
What is your goal? VGA and HDMI are only "image transfer cable". This will transfer or duplicate the image from your laptop or PC to the TV.
To build a network you would need a LAN cable from the laptop to the PC or alternatively it can also go via Wi-Fi, if your TV is a SmartTV.
This has nothing to do with a network. The only direct their video signal to the TV. If you want to build a real network, you need a switch where the devices are connected and s.besten a router.