A192027 - OEIS

%I #12 Mar 29 2020 13:08:38

%S 1,10,4,27,29,9,60,75,58,16,105,160,147,97,25,174,275,308,243,146,36,

%T 259,447,525,504,363,205,49,376,658,846,855,748,507,274,64,513,944,

%U 1239,1371,1265,1040,675,353,81,690,1278,1768,2002,2022,1755,1380,867,442,100

%N 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.

%C W(1,m) = m^2 = A000290(m).

%C W(2,m) = A079273(m+1).

%C W(n,1) = A180574(n).

%H B. E. Sagan, Y-N. Yeh and P. Zhang, <a href="http://users.math.msu.edu/users/sagan/Papers/Old/wpg-pub.pdf">The Wiener Polynomial of a Graph</a>, Internat. J. of Quantum Chem., 60, 1996, 959-969.

%F 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;

%F 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.

%F 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.

%e 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.

%e The square array starts:

%e    1,   4,   9,  16,  25,   36,   49, ...;

%e   10,  29,  58,  97, 146,  205,  274, ...;

%e   27,  75, 147, 243, 363,  507,  675, ...;

%e   60, 160, 308, 504, 748, 1040, 1380, ...;

%p 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

%p 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 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);

%Y Cf. A000290, A079273, A180574.

%K nonn,tabl

%O 1,2

%A _Emeric Deutsch_, Jun 26 2011