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David Eppstein
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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). Another would be and the drawings of sociograms in

Moreno, J. L. (1934), Who Shall Survive?, New York, N.Y.: Beacon House.

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 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). Another would be the drawings of sociograms in

Moreno, J. L. (1934), Who Shall Survive?, New York, N.Y.: Beacon House.

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.

Source Link
David Eppstein
  • 18.6k
  • 2
  • 55
  • 127

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 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). Another would be the drawings of sociograms in

Moreno, J. L. (1934), Who Shall Survive?, New York, N.Y.: Beacon House.