A143366 -id:A143366 - cp's OEIS frontend

A214580 The hyper-Wiener index of the circumcoronene H(n) (see definition in the Klavzar papers).

42, 2697, 29805, 163914, 616008, 1819539, 4550763, 10075380, 20321478, 38078781, 67224201, 112973694, 182160420, 283539207, 428117319, 629511528, 904331490, 1272589425, 1758136101, 2389123122, 3198491520, 4224486651, 5511199395, 7109133660, 9075800190, 11476336677, 14384154177

Offset: 1

Emeric Deutsch, Aug 31 2012

The hyper-Wiener index of a connected graph is (1/2)*Sum [d(i,j)+d(i,j)^2], where d(i,j) is the distance between the vertices i and j and summation is over all unordered pairs of vertices (i,j).
The Wiener index is in A143366.
The Wiener polynomials for n=1,2,3,4,5 are given in A214581.

			a(1)=42: for n=1 we have a hexagon; the distances are: 1 (6 times), 2 (6 times), 3 (3 times). Then a(1)=(1/2)*(6*1+6*2+3*3+6*1+6*4+3*9)=42.

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.
P. Zigert, S. Klavzar, and I. Gutman, Calculating the hyper-Wiener index of benzenoid hydrocarbons, ACH Models Chem., 137, 2000, 83-94.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A143366, A214581.

a := proc (n) options operator, arrow: (1/10)*n+(17/15)*n^2-3*n^3-(55/6)*n^4+(82/5)*n^5+(548/15)*n^6 end proc: seq(a(n), n = 1 .. 30);

LinearRecurrence[{7,-21,35,-35,21,-7,1},{42,2697,29805,163914,616008,1819539,4550763},30] (* Harvey P. Dale, Feb 11 2024 *)

a(n) = (1/10)n +(17/15)n^2 -3n^3 -(55/6)n^4 +(82/5)n^5 +(548/15)n^6.

G.f. = 3*x*(14 +801*x +3936*x^2 +3482*x^3 +530*x^4 +5*x^5)/(1-x)^7.

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).

Original entry on oeis.org

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

Emeric Deutsch, Aug 31 2012

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).

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.

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

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).

cp's OEIS Frontend

A214580 The hyper-Wiener index of the circumcoronene H(n) (see definition in the Klavzar papers).

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