<p>The Lipschitz space of an infinite (locally finite) graph is defined as the set of functions on the vertices of the graph such that the differences of the values between adjacent vertices remain bounded. In this paper, we prove that this set is a Banach space when endowed with its natural norm, and we define the little Lipschitz space as the subspace where these differences tend to zero. We consider the multiplication operators on these spaces and characterize their boundedness, compactness, and the spectra. We also obtain estimates of the norm and essential norm, and we characterize when these operators are isometric.</p>