**Core Concept**
The external carotid artery supplies blood to the face, including the nasal cavity. Ligation of this artery may not completely eliminate the risk of epistaxis (nosebleed) due to the presence of other blood supply routes.

**Why the Correct Answer is Right**
The ophthalmic artery, a branch of the internal carotid artery, provides a significant portion of the blood supply to the nasal cavity. After ligation of the external carotid artery, the ophthalmic artery may become the primary source of blood flow to the nasal mucosa, increasing the risk of epistaxis. This is particularly relevant in cases where the ophthalmic artery is large or well-developed.

**Why Each Wrong Option is Incorrect**
**Option A:** The maxillary artery, another branch of the external carotid artery, is not primarily responsible for supplying the nasal cavity, making it an unlikely cause of epistaxis after ligation of the external carotid artery.
**Option B:** The internal maxillary artery, a branch of the external carotid artery, does contribute to the blood supply of the nasal cavity but is not the primary source after ligation of the external carotid artery.
**Option C:** The sphenopalatine artery, a branch of the maxillary artery, is involved in the blood supply of the nasal cavity but is not the primary source after ligation of the external carotid artery.

**Clinical Pearl / High-Yield Fact**
It is essential to consider the possibility of ophthalmic artery-mediated epistaxis after ligation of the external carotid artery, particularly in cases where the nasal mucosa remains congested or bleeding persists despite initial management.

**Correct Answer: C. The ophthalmic artery.**