A228314 Triangle read by rows: T(p,q) (1<=q<=p) is the hyper-Wiener index of the Cartesian product of the cycles C(p) and C(q) (Torus Grid Graph).
0, 1, 10, 3, 27, 72, 10, 72, 186, 448, 20, 135, 345, 810, 1450, 42, 258, 648, 1464, 2580, 4482, 70, 413, 1029, 2282, 3990, 6846, 10388, 120, 672, 1656, 3584, 6200, 10464, 15736, 23552, 180, 981, 2403, 5130, 8820, 14742, 22050, 32760, 45360, 275, 1450, 3525, 7400, 12625
Offset: 1
Links
- B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.
Programs
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Maple
HWWi := proc (p, q) if `mod`(p, 2) = 1 and `mod`(q, 2) = 1 then (1/96)*p*q*(3*p^2*q^2+2*p^3*q+2*p*q^3-4*p*q-3*p^2-3*q^2-6*p-6*q+6*p^2*q+6*p*q^2+3) elif `mod`(p, 2) = 0 and `mod`(q, 2) = 0 then (1/96)*p^2*q^2*(6*q+6*p+3*p*q+2*p^2+2*q^2+8) elif `mod`(p, 2) = 1 and `mod`(q, 2) = 0 then (1/96)*p*q^2*(3*p^2*q+2*p^3+2*p*q^2+2*p-3*q-6+6*p^2+6*p*q) else (1/96)*p^2*q*(3*p*q^2+2*q^3+2*p^2*q+2*q-3*p-6+6*q^2+6*p*q) end if end proc: for i to 10 do seq(HWWi(i, j), j = 1 .. i) end do; # yields sequence in triangular form H := proc (p, q) local br, h: br := proc (n) options operator, arrow: sum(t^k, k = 0 .. n-1) end proc; h := proc (m) if `mod`(m, 2) = 0 then m*(br((1/2)*m)-1)+(1/2)*m*t^((1/2)*m) else m*t*br((1/2)*m-1/2) end if end proc: sort(expand(2*h(p)*h(q)+p*h(q)+q*h(p))) end proc: Wi := proc (p, q) options operator, arrow: subs(t = 1, diff(H(p, q), t)) end proc: for i to 10 do seq(Wi(i, j), j = 1 .. i) end do; # yields sequence in triangular form
Formula
T(p,q) = pq(3-4pq-6p-6q-3p^2-3q^2+6pq^2+6p^2q+2pq^3+3p^2q^2+2p^3q)/96 if both p and q are odd.
T(p,q) = p^2*q^2*(8+6p + 6q+2p^2+3pq+2q^2)/96 if both p and q are even.
T(p,q) = pq^2*(2p-3q-6+6pq+ 6p^2 +2pq^2+3p^2q+2p^3)/96 if p is odd and q is even.
T(p,q) = p^2*q*(2q-3p-6+6pq+6q^2+2qp^2+3pq^2+2q^3)/96 if p is even and q is odd.
The first Maple program makes use of the above formulas.
The Hosoya-Wiener polynomial of C(p) X C(q) is 2*h(p)*h(q) + p*h(q) + q*h(p), where h(j) denotes the Hosoya-Wiener polynomial of the cycle C(j).
The command H(p,q) in the 2nd Maple program yields the Hosoya-Wiener polynomial.
Comments