A214581 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the circumcoronene H(n) (n=1,2,3,4,5; see definition in the Klavzar papers).

6, 6, 3, 30, 48, 57, 54, 45, 30, 12, 72, 126, 165, 186, 195, 186, 168, 138, 102, 66, 27, 132, 240, 327, 390, 435, 456, 462, 444, 414, 366, 309, 246, 177, 114, 48, 210, 390, 543, 666, 765, 834, 882, 900, 900, 870, 825, 756, 675, 582, 480, 378, 270, 174, 75

Offset: 1

Author

Emeric Deutsch, Aug 31 2012

Keywords

• nonn
• tabf
• more

Comments

The entries in row n are the coefficients of the Wiener polynomial of the corresponding graph.

Row n contains 4n-1 entries.

T(n,1) = 9n^2-3n = A152743(n).

T(n,2) = 6n(3n-2)= A153796(n).

T(n,3) = 3(9n^2-9n+1)= 3*A069131(n) (for n>5 this is a conjecture).

T(n,2n) = n(7n^2-1) = 6*A004126(n) (for n>5 this is a conjecture).

T(n,4n-2) = 6(n^2+n-1) = 6*A028387(n-1) (for n>5 this is a conjecture).

T(n,4n-1) = 3n^2 = A033428(n) (for n>5 this is a conjecture).

Sum(k*T(n,k), k>=1) = A143366(n).

Links

• S. Klavzar, A bird's eye view of the cut method and a survey of its applications in chemical graph theory, MATCH, Commun. Math. Comput. Chem. 60, 2008, 255-274.
• Bo-Yin Yang and Yeong-Nan Yeh, A Crowning Moment for Wiener Indices, Studies in Appl. Math., 112 (2004), 333-340.
• Bo-Yin Yang and Yeong-Nan Yeh, Wiener polynomials of some chemically interesting graphs, International Journal of Quantum Chemistry, 99 (2004), 80-91, 2004.
• P. Zigert, S. Klavzar, and I. Gutman, Calculating the hyper-Wiener index of benzenoid hydrocarbons, ACH Models Chem., 137, 2000, 83-94.

Crossrefs

Cf. A143366, A214580, A152743, A153796, A069131, A004126, A028387, A033428.

Formula

The entries have been obtained by using the Maple Graph Theory package for finding the distance matrix of each of the five graphs H(n) (n=1,2,3,4,5). The given Maple program yields the Wiener polynomial of H(2) (having as coefficients the entries in row 2).