Extremal Mono-q-polyhexes

A polyhex can be defined as a graph H which is obtained from the hexagonal lattice L6 of the plane by taking a cycle C in the graph of L6 and defining H to be the part of L6 in the disk bounded by C, including C. Similarly, a mono-qpolyhex, q an integer greater than 3, is obtained from the mono-q-hexagonal lattice (all faces are hexagons except one which is a q-gon, and three faces meet at each vertex) by specifying a cycle Csuch that the q-gon is included in the disk bounded by C. A simple description of the mono-q-hexagonal lattice is as follows. Let L1 be one-sixth of the hexagonal lattice obtained by taking one of its hexagons Q, the middle point x of Q, and the six rays from x through each vertex of Q toward infinity (see Figure 1). Then L1 is one of the wedges of the plane between the two consecutive rays. The mono-q-hexagonal lattice is then obtained by taking q copies of L1 and identifying their sides in circular order. Note that L3, L4, L5, and L6 can be realized in 3-space so that all the hexagons remain congruent (as an unbounded cone with its apex in the middle of the q-gon), while L7, LS, ... cannot be modeled with congruent hexagons due to their hyperbolic nature. The above description of L, also helps us to represent the mono-q-polyhexes by specifying the q wedges in the copies of L1 (see Figure 2). Let H by a mono-q-polyhex, and let nint and h denote the number of internal vertices of H and the number of hexagons (if q = 6, let h be the number of hexagons minus 1, so that the q-face is not counted by h) , respectively. A useful relation was derived by Gutman2 in case of q = 6: