The first batter in either the top or the bottom of an inning hits a home run. By definition, this is a one-run homer, because it is impossible for any runner to be on base when the first batter comes up.
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It is when the home team wins a game with a home run in the bottom of the ninth or in extra innings.
A walk-off home run is a home run that ends the game. It must be the run that gives the home team the lead in the final inning of the game.