So to be clear, you are connecting to your router through the SSID Carol? Which are the connections you are worried about?
You can see what they call MAC addresses for each device, the long alphanumeric numbers which are hard coded into each device. You can find these numbers on any of your devices to confirm what is what. If after you have gone round the house identifying all the connected devices MAC addresses against what is connected you still have mystery connections simply boot them off the network, anything you may has missed will soon scream out they are not connection. You should also change the wifi password so these devices can no longer connect to the network.
all i know is signal strength indicates its in the home. Utilities have been of for a month. I believe more girlfriend is lying and knows exactly what it is, but act's like she doesn't.