If you mean by italics the word encased in asterisks, that word is option number two: the Verb
In that particular sentence that you list,
Brian is the noun (as well as the subject),
Walked is the verb (as it describes the action of the clause or sentence.
slowly is the adverb, (as it qualifies the verb)
walked slowly home, as the day was so warm and pleasant is the predicate.
