Duality in geometric graphs: vector graphs, Kirchhoff graphs and Maxwell reciprocal figures (Q2412324)

Summary: We compare two mathematical theories that address duality between cycles and vertex-cuts of graphs in geometric settings. First, we propose a rigorous definition of a new type of graph, vector graphs. The special case of \(\mathbb{R}^2\)-vector graphs matches the intuitive notion of drawing graphs with edges taken as vectors. This leads to a discussion of Kirchhoff graphs, as originally presented by the third author [SIAM J. Appl. Math. 70, No. 2, 543--562 (2009; Zbl 1220.05126); Ars Math. Contemp. 9, No. 1, 125--144 (2015; Zbl 1329.05130)], which can be defined independent of any matrix relations. In particular, we present simple cases in which vector graphs are guaranteed to be Kirchhoff or non-Kirchhoff. Next, we review Maxwell's method of drawing reciprocal figures as he presented in 1864, using modern mathematical language. We then demonstrate cases in which \(\mathbb{R}^2\)-vector graphs defined from Maxwell reciprocals are ``dual'' Kirchhoff graphs. Given an example in which Maxwell's theories are not sufficient to define vector graphs, we begin to explore other methods of developing dual Kirchhoff graphs.