Who first used/gave a coordinate representation of a graph? In his proof of the Shannon capacity of a graph, Lovasz utilizes a coordinate representation of the pentagon (namely an orthonormal representation). Who first utilized a coordinate representation for finite/infinite graphs for any purpose? I am thinking a general vector valued representation. And not just a $2$-d representation or a 'fixed' dimensional representation. 
 A: If by "coordinate representation" you mean the assignment of geometric positions to a graph that was initially given as a purely combinatorial structure, then one possible contender is the proof of Steinitz's theorem by
Steinitz, E. (1922), "Polyeder und Raumeinteilungen", Encyclopädie der mathematischen Wissenschaften, Band 3 (Geometries), pp. 1–139.
Somewhat later we have the proof of Fáry's theorem by
Wagner, Klaus (1936), "Bemerkungen zum Vierfarbenproblem", Jahresbericht der Deutschen Mathematiker-Vereinigung 46: 26–32.
(Fáry's and Stein's independent discoveries were later) and the drawings of sociograms in
Moreno, J. L. (1934), Who Shall Survive?, New York, N.Y.: Beacon House.
A: Maybe not quite a graph, but perhaps close enough:
From Wikipedia: Ptolemy's world map is a map of the known world to Hellenistic society in the 2nd century CE. It was based on the description contained in Ptolemy's book Geographia, written c. 150. Although authentic maps have never been found, the Geographia contains thousands of references to various parts of the old world, with coordinates for many, allowing cartographers to reconstruct Ptolemy's world view around 1300 when the manuscript was re-discovered. According to Ptolemy's books, only the actual map of world was made through mathematical calculations.
Perhaps the most significant contributions of Ptolemy's maps are the first uses of longitudinal and latitudinal lines as well as specifying terrestrial locations by celestial observations. Geographia was translated from Greek into Arabic in the 9th century. The idea of a global coordinate system revolutionized medieval Islamic and European geographical thought, as it was based upon a scientific and numerical basis.
