Which condition is characterized by peripheral ipsilateral facial paralysis with an inability to close the eye on the involved side?

The condition characterized by peripheral ipsilateral facial paralysis and an inability to close the eye on the affected side is linked to dysfunction of cranial nerve VII, which is responsible for controlling the muscles of facial expression. When there is an issue with this nerve, such as in Bell's palsy or other forms of facial nerve damage, it leads to weakness or paralysis on one side of the face. This dysfunction typically manifests as drooping of the mouth, inability to raise the eyebrow, and an inability to close the eye, highlighting the facial nerve's crucial role in facial movements.

The other options represent different cranial nerves that do not primarily function in facial expression or eye closure. Dysfunction of cranial nerve I relates to the sense of smell, cranial nerve XII is responsible for motor function of the tongue, and cranial nerve VI controls lateral eye movement. Therefore, these nerves would not account for the specific symptoms of facial paralysis and inability to close the eye, making cranial nerve VII the clear choice for this condition.