Duality in geometric graphs: vector graphs, Kirchhoff graphs and Maxwell reciprocal figures. (English) Zbl 1420.05044

Reese, Tyler; Paffenroth, Randy; Fehribach, Joseph D.

Symmetry 8, No. 3, Article ID 9, 28 p. (2016).

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.

MSC:

05C10

Planar graphs; geometric and topological aspects of graph theory

Keywords:

Kirchhoff graphs; geometric graphs; duality; Maxwell reciprocals

Citations:

Zbl 1220.05126; Zbl 1329.05130