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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.

%I A192026 #11 Apr 02 2017 03:19:15
%S A192026 36,72,90,120,180,168,180,300,336,270,252,450,560,540,396,336,630,840,
%T A192026 900,792,546,432,840,1176,1350,1320,1092,720,540,1080,1568,1890,1980,
%U A192026 1820,1440,918,660,1350,2016,2520,2772,2730,2400,1836,1140,792,1650,2520,3240,3696,3822,3600,3060,2280,1386
%N 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.
%H A192026 B. E. Sagan, Y-N. Yeh and P. Zhang, <a href="http://dx.doi.org/10.1002/(SICI)1097-461X(1996)60:5&lt;959::AID-QUA2&gt;3.0.CO;2-W">The Wiener Polynomial of a Graph</a>, Internat. J. of Quantum Chem., 60, 1996, 959-969.
%F A192026 W(n,1) = A049598(n-1).
%F A192026 W(n,m) = n*(n-1)*(m+1)*(2*m+1) (n >= 3, m >= 1).
%F A192026 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.
%e A192026 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.
%e A192026 The square array starts:
%e A192026    36,  90, 168,  270,  396,  546,  720,  918, ...;
%e A192026    72, 180, 336,  540,  792, 1092, 1440, 1836, ...;
%e A192026   120, 300, 560,  900, 1320, 1820, 2400, 3060, ...;
%e A192026   180, 450, 840, 1350, 1980, 2730, 3600, 4590, ...;
%p A192026 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
%p A192026 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.
%Y A192026 Cf. A049598.
%K A192026 nonn,tabl
%O A192026 3,1
%A A192026 _Emeric Deutsch_, Jun 25 2011