Adjacency Matrices and Their Eigenvalues for Toroidal Polyhexes

Adjacency Matrices and Their Eigenvalues for Toroidal Polyhexes 

Three cases have been distinguished, according to the numerical properties of a, b, d and the nature of the cyclic pattern of hexagons on the torus. 

Structures with up to 7200 vertices (3600 hexagons) can be dealt with under Cases I and II. Beyond this, certain structures come under Case HI only, which is more complicated to deal with. Specifically, Case I can be used if and only if b is divisible by the highest common factor of a and d. It covers all polyhexes with fewer than 1800 vertices and 900 hexagons (and many, although not all, larger ones). It can be proved that at least one of the matrices equivalent to (a-b-d) has this required property unless ad is divisible by the squares of three different primes, but often it can even then, i.e., this prime number condition is necessary but not sufficient for exclusion from consideration under this Case. 

The smallest number that is divisible by the squares of three distinct prime numbers is 900 (equivalent to 2 x 3 x 5 ), and TPH(450-5-2) is an example of a toroidal polyhex that must be brought under a higher Case. 

Case II requires that there must be no integer greater than one that divides all of a, b, d the case is therefore a useful supplement or alternative to Case I and, together with Case I, allows all structures with up to 7200 vertices (3600 hexagons) to be dealt with. 

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