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

Author

Emeric Deutsch, Aug 26 2013

Keywords

• nonn
• tabl

Comments

T(n,1) = A213850(n-1).

T(n,n) = A228316(n).

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.

Crossrefs

Cf. A213850, A228313, A228316.

Programs

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.