Multi-Layered Cyclic Fence Graphs. Discovery of New Series of Graphs with Exceedingly High Symmetry

Abstract

Two series of multi-layered cyclic fence graphs, E_{m n} and F_{m n} are proposed to be defined, both of which are composed of m 2n-membered cycle graphs with periodic bridging and are all cubic (degree of all the vertices being three) and bipartite. These new series of graphs generally have exceedingly high symmetry. Especially, a few members of them have such a property that all the edges and vertices are respectively equivalent, i.e., the edge and vertex topicities are all unity. All the E and F families can be mapped on a torus in such a manner that they are composed of only hexagons and each hexagon is surrounded by six hexagons. The analytical form of the solutions of the characteristic polynomial of these graphs were obtained. Thus one can correctly estimate the electronic structure of the infinitely large graphite network from the analysis of the selected series of the newly proposed graphs, E and F.