Learn Russian: Окончив университет, выпускник нашёл работу. - Having graduated from the university, the graduate found a job.

It is a perfective verbal adverb, meaning it describes a completed action that happened before the main verb. To form it, take the past tense of the perfective verb окончить (which is окончил), drop the -л, and add the suffix -в. This gives us окончив (having graduated).

Could I use this structure if someone else found the job for the graduate?

No, you cannot. In Russian, the action of the verbal adverb (окончив) and the action of the main verb (нашёл) must be performed by the exact same subject (выпускник). If the subjects are different, you must use a complex sentence with words like после того как (after).

Why does университет stay in the accusative case after окончив?

Verbal adverbs keep the same case requirements as the original verb. Because the verb окончить takes a direct object in the accusative case, its verbal adverb окончив also requires the accusative case for университет.

Why use окончив instead of закончив?

Both come from verbs meaning to finish. However, окончить is the specific verb traditionally used for graduating from an educational institution like a university or school. Закончить is generally used for finishing everyday tasks or projects.