A192026 Square array read by antidiagonals: W(n,m) (n >= 3, m >= 1) is the Wiener index of the graph G(n,m) obtained from an n-wheel graph by adjoining m pendant edges at each node of the cycle.

36, 72, 90, 120, 180, 168, 180, 300, 336, 270, 252, 450, 560, 540, 396, 336, 630, 840, 900, 792, 546, 432, 840, 1176, 1350, 1320, 1092, 720, 540, 1080, 1568, 1890, 1980, 1820, 1440, 918, 660, 1350, 2016, 2520, 2772, 2730, 2400, 1836, 1140, 792, 1650, 2520, 3240, 3696, 3822, 3600, 3060, 2280, 1386

Offset: 3

Author

Emeric Deutsch, Jun 25 2011

Keywords

• nonn
• tabl

Examples

			W(3,1)=36 because in the graph with vertex set {O,A,B,C,A',B',C'} and edge set {OA, OB, OC, AB, BC, CA, AA', BB', CC'} we have 9 pairs of vertices at distance 1 (the edges), 9 pairs at distance 2 (A'O, A'B, A'C, B'O, B'A, B'C, C'O, C'A, C'B) and 3 pairs at distance 3 (A'B', B'C', C'A'); 9*1 + 9*2 + 3*3 = 36.
The square array starts:
   36,  90, 168,  270,  396,  546,  720,  918, ...;
   72, 180, 336,  540,  792, 1092, 1440, 1836, ...;
  120, 300, 560,  900, 1320, 1820, 2400, 3060, ...;
  180, 450, 840, 1350, 1980, 2730, 3600, 4590, ...;
		

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. A049598.

Programs

Maple

W := proc (n, m) options operator, arrow: n*(n-1)*(m+1)*(2*m+1) end proc: for n from 3 to 12 do seq(W(n-i, i+1), i = 0 .. n-3) end do; # yields the antidiagonals in triangular form
W := proc (n, m) options operator, arrow: n*(n-1)*(m+1)*(2*m+1) end proc: for n from 3 to 12 do seq(W(n, m), m = 1 .. 10) end do; # yields the first 10 entries of each of rows 3,4,...,12.

Formula

W(n,1) = A049598(n-1).

W(n,m) = n*(n-1)*(m+1)*(2*m+1) (n >= 3, m >= 1).

The Wiener polynomial of the graph G(n,m) is P(n,m;t) = n*(m+2)*t + (1/2)*n*(m^2+n+5*m-3)*t^2 + n*m*(m+n-3)*t^3 + (1/2)*n*m^2*(n-3)*t^4.