On the girth of three-dimensional algebraically defined graphs with multiplicatively separable functions (Q2094893)

A three-dimensional algebraically defined graph \({\Gamma}_{\mathcal{R}}(f_2(X,Y), f_3(X, Y))\), for a ring \(\mathcal{R}\) and functions \(f_2,f_3: \mathcal{R}^2 \to \mathcal{R}\) is a bipartite graph where each partite set is a copy of \(\mathcal{R}^3\) and the functions \(f_2\) and \(f_3\) are used for the edges. There are different results and open questions on these types of graphs, like isomorphism questions, connectivity, and their girth. The paper under review is concerned with the results on the girth of these graphs when \(f_2(X, Y) = f(X)h(Y)\) and \(f_3(X, Y) = g(X)j(Y)\) and \(\mathcal{R} = \mathbb{F}\) is a field. The results are on the classification of these types of graphs using the possible values of the girth. First, the results are given in the case when \(h = j\) and when \(\mathbb{F} = \mathbb{R}\). The methods used extend when the field of reals is replaced by any ordered field. In the next step, the results are extended, with suitable assumptions, to the general case. The analogues of the results are given when \(\mathbb{F}\) is a finite field. Applications of the results over the reals are given, when specific, such as exponentials, radicals, and trigonometric, functions are used. In the end, the authors state the cases that are left open from their approach.