A180574 -id:A180574 - cp's OEIS frontend

A180573 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in the sun graph on 2n nodes. The sun graph on 2n nodes is obtained by attaching n pendant edges to the cycle graph on n nodes.

6, 6, 3, 8, 10, 8, 2, 10, 15, 15, 5, 12, 18, 21, 12, 3, 14, 21, 28, 21, 7, 16, 24, 32, 28, 16, 4, 18, 27, 36, 36, 27, 9, 20, 30, 40, 40, 35, 20, 5, 22, 33, 44, 44, 44, 33, 11, 24, 36, 48, 48, 48, 42, 24, 6, 26, 39, 52, 52, 52, 52, 39, 13, 28, 42, 56, 56, 56, 56, 49, 28, 7, 30, 45, 60

Offset: 3

Emeric Deutsch, Sep 19 2010

Number of entries in row n = 2 + floor(n/2).
Sum of entries in row n = n(2n-1)=A000384(n).
Sum(k*T(n,k),k>=1) = A180574(n).

			Triangle starts:
6,6,3;
8,10,8,2;
10,15,15,5;
12,18,21,12,3;

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.
D. Stevanovic, Hosoya polynomial of composite graphs, Discrete Math., 235 (2001), 237-244.

Cf. A000384, A180574

P := proc (n) if `mod`(n, 2) = 0 then sort(expand(n*t*(1+t)^2*(sum(t^j, j = 0 .. (1/2)*n-1))+n*t-(1/2)*n*t^((1/2)*n)*(1+t)^2)) else sort(expand(n*t*(1+(1+t)^2*(sum(t^j, j = 0 .. (1/2)*n-3/2))))) end if end proc; for n from 3 to 15 do P(n) end do: for n from 3 to 15 do seq(coeff(P(n), t, i), i = 1 .. 2+floor((1/2)*n)) end do; # yields sequence in trianguklar form

The generating polynomial of row 2n is 2nt-nt^n*(1+t)^2+2nt(1+t)^2*sum(t^j, j=0..n-1); the generating polynomial of row 2n+1 is (2n+1)t[1+(1+t)^2*sum(t^j,j=0..n-1)]; these are the Wiener polynomials of the corresponding graphs.

A192027 Square array read by antidiagonals: W(n,m) (n >= 1, m >= 1) is the Wiener index of the graph G(n,m) obtained from the n-circuit graph by adjoining m pendant edges at each node of the circuit.

Original entry on oeis.org

1, 10, 4, 27, 29, 9, 60, 75, 58, 16, 105, 160, 147, 97, 25, 174, 275, 308, 243, 146, 36, 259, 447, 525, 504, 363, 205, 49, 376, 658, 846, 855, 748, 507, 274, 64, 513, 944, 1239, 1371, 1265, 1040, 675, 353, 81, 690, 1278, 1768, 2002, 2022, 1755, 1380, 867, 442, 100

Offset: 1

Emeric Deutsch, Jun 26 2011

W(1,m) = m^2 = A000290(m).
W(2,m) = A079273(m+1).
W(n,1) = A180574(n).

			a(3,1)=27 because in the graph with vertex set {A,B,C,A',B',C'} and edge set {AB, BC, CA, AA', BB', CC'} we have 6 pairs of vertices at distance 1 (the edges), 6 pairs at distance 2 (A'B, A'C, B'A, B'C, C'A, C'B) and 3 pairs at distance 3 (A'B', B'C', C'A'); 6*1 + 6*2 + 3*3 = 27.
The square array starts:
   1,   4,   9,  16,  25,   36,   49, ...;
  10,  29,  58,  97, 146,  205,  274, ...;
  27,  75, 147, 243, 363,  507,  675, ...;
  60, 160, 308, 504, 748, 1040, 1380, ...;

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.

Cf. A000290, A079273, A180574.

W := proc (n, m) if `mod`(n, 2) = 0 then (1/2)*n*((1/4)*n^2+2*m^2*n+(1/4)*m^2*n^2+2*m*n+(1/2)*m*n^2-2*m) else (1/8)*(n^2-1+m^2*n^2+8*m^2*n-m^2+2*m*n^2+8*m*n-10*m)*n end if end proc: for n to 10 do seq(W(n-i, i+1), i = 0 .. n-1) end do; # yields the antidiagonals in triangular form
W := proc (n, m) if `mod`(n, 2) = 0 then (1/2)*n*((1/4)*n^2+2*m^2*n+(1/4)*m^2*n^2+2*m*n+(1/2)*m*n^2-2*m) else (1/8)*(n^2-1+m^2*n^2+8*m^2*n-m^2+2*m*n^2+8*m*n-10*m)*n end if end proc: for n to 10 do seq(W(n, m), m = 1 .. 10) end do; # yields the first 10 entries of each of rows 1,2,...,10.
P := proc (n, m) if `mod`(n, 2) = 0 then sort(expand(n*(m*t+(1/2)*m*(m-1)*t^2)+n*(sum(t^j, j = 1 .. (1/2)*n-1))*(1+m*t)^2+(1/2)*n*t^((1/2)*n)*(1+m*t)^2)) else sort(expand(n*(m*t+(1/2)*m*(m-1)*t^2)+n*(sum(t^j, j = 1 .. (1/2)*n-1/2))*(1+m*t)^2)) end if end proc: P(4,9);

If n even, then: W(n,m) = n*(n^2/4 + 2*m^2*n + m^2*n^2/4 + 2*m*n + m*n^2/2 - 2*m)/2;
if n odd, then: W(n,m) = n*(n^2 - 1 + m^2*n^2 + 8*m^2*n - m^2 + 2*m*n^2 + 8*m*n - 10*m)/8.
The Wiener polynomial P(n,m;t) of the graph G(n,m) is given in the 3rd Maple program. It gives, for example, P(4,9) = 162*t^4 + 360*t^3 + 218*t^2 + 40*t. Its derivative, evaluated at t=1, yields the corresponding Wiener index W(4,9)=4184.

A192028 Square array read by antidiagonals: W(n,m) (n >= 1, m >= 1) is the Wiener index of the graph G(n,m) obtained from the n-circuit graph by joining at each of its nodes a path with m nodes (n >= 1, m >= 1; if m=1, then the n-circuit is not modified).

Original entry on oeis.org

0, 1, 1, 3, 10, 4, 8, 27, 35, 10, 15, 60, 93, 84, 20, 27, 105, 196, 222, 165, 35, 42, 174, 335, 456, 435, 286, 56, 64, 259, 537, 770, 880, 753, 455, 84, 90, 376, 784, 1212, 1475, 1508, 1197, 680, 120, 125, 513, 1112, 1750, 2295, 2515, 2380, 1788, 969, 165

Offset: 1

Emeric Deutsch, Jun 27 2011

W(1,m) = A000292(m-1).
W(2,m) = A000447(m) = A000292(2m-2).
W(n,1) = A034828(n).
W(n,2) = A180574(n) (n >= 3).

			a(3,1)=27 because in the graph with vertex set {A,B,C,A',B',C'} and edge set {AB, BC, CA, AA', BB', CC'} we have 6 pairs of vertices at distance 1 (the edges), 6 pairs at distance 2 (A'B, A'C, B'A, B'C, C'A, C'B) and 3 pairs at distance 3 (A'B', B'C', C'A'); 6*1 + 6*2 + 3*3 = 27.
The square array starts:
  0,  1,   4,  10,  20,   35,   56,   84, ...;
  1, 10,  35,  84, 165,  286,  455,  680, ...;
  3, 27,  93, 222, 435,  753, 1197, 1788, ...;
  8, 60, 196, 456, 880, 1508, 2380, 3536, ...;

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.

Cf. A000292, A000447, A034828, A180574.

W := proc (n, m) if `mod`(n, 2) = 0 then (1/24)*n*m*(3*n^2*m+12*n*m^2-8*m^2-12*n*m+12*m-4) else (1/24)*n*m*(3*n^2*m+12*n*m^2-8*m^2-12*n*m+9*m-4) end if end proc: for n to 10 do seq(W(n-i, i+1), i = 0 .. n-1) end do; # yields the antidiagonals in triangular form
W := proc (n, m) if `mod`(n, 2) = 0 then (1/24)*n*m*(3*n^2*m+12*n*m^2-8*m^2-12*n*m+12*m-4) else (1/24)*n*m*(3*n^2*m+12*n*m^2-8*m^2-12*n*m+9*m-4) end if end proc: for n to 10 do seq(W(n, m), m = 1 .. 10) end do; # yields the first 10 entries of each of rows 1,2,...,10.
P := proc (n, m) if `mod`(n, 2) = 0 then sort(expand(simplify(n*t*(t^m-m*t+m-1)/(1-t)^2+(n*(sum(t^j, j = 1 .. (1/2)*n-1))+(1/2)*n*t^((1/2)*n))*(1-t^m)^2/(1-t)^2))) else sort(expand(simplify(n*t*(t^m-m*t+m-1)/(1-t)^2+n*(sum(t^j, j = 1 .. (1/2)*n-1/2))*(1-t^m)^2/(1-t)^2))) end if end proc: P(3, 4);

W(n,m) = (1/24)*n*m*(3*m*n^2 + 12*n*m^2 - 8*m^2 - 12*n*m + 12*m - 4) if n is even;
W(n,m) = (1/24)*n*m*(3*m*n^2 + 12*n*m^2 - 8*m^2 - 12*n*m + 9*m - 4) if n is odd.
The Wiener polynomial P(n,m;t) of the graph G(n,m) is given in the 3rd Maple program. It gives, for example, P(3,4) = 12*t + 12*t^2 + 12*t^3 + 12*t^4 + 9*t^5 + 6*t^6 + 3*t^7. Its derivative, evaluated at t=1, yields the corresponding Wiener index W(3,4)=222.

cp's OEIS Frontend

A180573 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in the sun graph on 2n nodes. The sun graph on 2n nodes is obtained by attaching n pendant edges to the cycle graph on n nodes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A192027 Square array read by antidiagonals: W(n,m) (n >= 1, m >= 1) is the Wiener index of the graph G(n,m) obtained from the n-circuit graph by adjoining m pendant edges at each node of the circuit.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A192028 Square array read by antidiagonals: W(n,m) (n >= 1, m >= 1) is the Wiener index of the graph G(n,m) obtained from the n-circuit graph by joining at each of its nodes a path with m nodes (n >= 1, m >= 1; if m=1, then the n-circuit is not modified).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula