Final answer:
The nerve that runs with the superficial temporal artery and vein anterior to the ear is linked with branches of the facial nerve. The superficial temporal artery, important for its relation here, is a terminal branch of the external carotid artery.
Step-by-step explanation:
The nerve that runs with the superficial temporal artery and vein anterior to the ear is part of the branches of the facial nerve (cranial nerve VII). The superficial temporal artery is a terminal branch of the external carotid artery which arises from the common carotid artery. The common carotid artery itself is closely associated with the internal carotid artery that after passing through the carotid canal of the temporal bone, enters the base of the brain and supplies it with blood through its branches like the middle cerebral artery that supplies blood to the temporal and parietal lobes of the cerebrum.