Eigenspectra and characteristic polynomials of cylindrical polyacene strips

Eigenspectra of cylindrical non-Möbius strips containing Z fused hexagons have been obtained in the analytical form where Z is an integer (≥ 2) and j = 1,2,3,…, Z. Further it has been shown that if Z = 2 np where p = 3, 4 or 5 and n is an integer, then the eigenvalues of the Z-ring cylindrical strip are given by where the number of surd signs (√) required within the square bracket is n; here r 3, r 4 and r 5 are the well known eigenvalues of the cycles C3, C4 and C5 respectively. Characteristic polynomials (CP) n of strips containing 2 n rings have been shown to obey the recurrence relation where f(x) is the CP of the linear chain with 2 n-1 vertices whose weights alternate as 0,1,0,1,… and the last edge carries a weight of √2. It has been shown also that (x 2 − x − 4) and (x 2 + x −4) are two common factors of the CPs of all cylindrical strips, irrespective of whether Z is even or odd, and in the case of even-Z strips x 2 (x 2 −1) is another common factor.