The Genus Distribution of Doubly Hexagonal Chains

By using the Transfer Matrix Method, explicit formulas for the embedding distribution of doubly hexagonal chain graphs are computed. Dedicated to Ante Graovac 1 Fasciagraphs and rotagraphs An interesting topic that I have learned from Ante Graovac in the 1980’s [1, 2] is that about fasciagraphs and rotagraphs. Such graphs are frequently studied in crystallography and in mathematical chemistry. They can be described by a small structure that is repeated as long chain that may be either open (fasciagraphs) or closed (rotagraphs). An important tool for dealing with these graphs is a rather general technique based on the Transfer Matrix Method . In addition to the afore-mentioned articles [1, 2], we refer to [18, 21] for some early applications of this method and to [17] for a more recent treatment in theoretical physics. Quite recently, the Transfer Matrix Method has been shown to be applicable to the problem of computing genus distributions of fasciagraphs and rotagraphs [15]. In this short note we will illustrate the method on a specific example of fasciagraphs that are isomorphic to the doubly hexagonal chain Dn of length n, which is shown in Figure 1 for n = 7. ∗Supported in part by an NSERC Discovery Grant, by the Canada Research Chair Program, and by the ARRS, Research Program P1-0297. 1 ar X iv :1 50 5. 03 66 4v 1 [ m at h. C O ] 1 4 M ay 2 01 5