Since Euler's result on the Seven Bridges of Königsberg problem appears to be the first known graph theoretical result I was wondering if any mathematicians after Euler and before Hamilton commented on it or investigated any other similar results. Perhaps prominent mathematicians like Lagrange, Laplace or Gauss.
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Yes, although Euler's work initially got so little attention that when Hierholzer proved existence of an Eulerian path in a graph with none or two vertices of odd degree (published posthumously in 1873) he was not aware of it. Euler himself stated the result without proof, only proving the converse. For the background on the original problem see What was the origin of the Seven Bridges of Königsberg problem before Euler? and the history section of Routledge's Handbook of Graph Theory. A comprehensive reference is 200 years of graph theory by Wilson.
The aftermath is described in detail in Graph Theory, 1736-1936 by Biggs, Lloyd and Wilson, ch.1, with translated excerpts from the original sources. The Königsberg's remembrance is especially endearing:
"Fortunately, Euler's work had not been completely forgotten, for in 1851 a French translation [2] by E. Coupy was published in the Nouvelles Annales de Mathematiques, a journal intended primarily for the students at the Ecole Polytechnique in Paris. Coupy also applied the methods of Euler to the analogous problem of the bridges over the River Seine. Nor was the problem forgotten in Königsberg; in 1875. L. Saalschütz [3] reported that a new bridge had been constructed there, joining the land areas denoted by B and C, and that the citizens' perambulation was now theoretically possible.
Coupy's interest in the work of Euler stemmed from two references to it in the early nineteenth century. The first of these occurred in a textbook of algebra [4], written by the Swiss mathematician Simon Antoine-Jean Lhuilier in 1804, and the second appeared in a memoir on polygons and polyhedra [5], written by Louis Poinsot in 1809... Poinsot proved, using arguments similar to those of Euler, that an Eulerian path in $K_n$ [complete graph on $n$ vertices] is impossible when $n = 4, 6, 8, ...$, because in these cases there are more than two vertices with odd valency. He also gave an ingenious method for constructing an Eulerian path in $K_n$ when $n$ is odd."
Poinsot was interested in solving diagram-tracing puzzles, i.e. tracing a graph "in a simple continuous stroke". It is possible if and only if the graph has an Eulerian path. Clausen in De linearum tertii ordinis proprietatibus (1844) and Listing were interested in them too. The latter even included them in his book Vorstudien zur Topologie (1847), which was Hierholzer's source and the first appearance of "topology" in print (Poincare will still call the subject by Leibniz's name Analysis Situs later into the century).
But Listing mentioned neither Euler nor Poinsot. Ironically, Hierholzer called Listing's treatise "unfortunately little-known" as well, but credited Listing with "incomplete proof" of his result. A commentary on Poinsot's memoir penned by Terquem appeared in the Nouvelles Annales de Mathematiques (1849), and included an interpretation of Poinsot's Eulerian path in $K_7$ as a complete game of dominoes. A general count of such games was published by Reiss in 1871, and a general count of Eulerian paths by Tarry in 1886. But the obliviousness continued, lengthy Wilson's On the traversing of geometrical figures (1905), published in Oxford, shows no awareness of any of the above. The subject is prone to independent rediscovery.