A180860 - OEIS

A180860

Square array read by antidiagonals: T(m,n) is the Wiener index of the tadpole graph L(m,n) (m>=3, n>=1). L(m,n) is the graph obtained by joining with an edge a node in the cycle graph C_m to an end-node of the path graph P_n. The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.

8, 16, 17, 26, 29, 31, 42, 43, 48, 51, 61, 64, 67, 74, 78, 88, 88, 94, 99, 108, 113, 119, 121, 124, 133, 140, 151, 157, 160, 158, 164, 170, 182, 191, 204, 211, 206, 206, 208, 218, 227, 242, 253, 268, 276, 264, 259, 264, 270, 284, 296, 314, 327, 344, 353, 328

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

3,1

COMMENTS

Tadpole graphs are also called keys (see the Gross & Yellen reference, p. 895).

T(m,1)=A180861(m).

REFERENCES

J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004.

LINKS

Table of n, a(n) for n=3..58.

Eric Weisstein's World of Mathematics, Tadpole Graph.

FORMULA

T(2m,n)=m^3+(1/6)n(n^2-1)+mn(m+n+1).

T(2m+1,n)=(1/2)m(m+1)(2m+1)+(1/6)n(n+1)(n+2)+mn(m+n+2).

EXAMPLE

T(3,1)=8 because the graph consists of a triangle ABCA and an edge AD; the distances are d(A,B)=d(B,C)=d(C,A)=d(A,D)=1 and d(DB)=d(DC)=2.

Square array T(i,j) begins:

8,17,31,51,78,...

16,29,48,74,108,...

26,43,67,99,140,...

42,64,94,133,182,...

MAPLE

T := proc (m, n) if `mod`(m, 2) = 0 then (1/8)*m^3+(1/6)*n*(n^2-1)+(1/4)*m*n*(m+2*n+2) else (1/8)*m*(m^2-1)+(1/6)*n*(n+1)*(n+2)+(1/4)*n*(m-1)*(m+2*n+3) end if end proc: for n from 3 to 13 do seq(T(n+1-i, i), i = 1 .. n-2) end do; # yields sequence in triangular form

CROSSREFS

Cf. A180861

Sequence in context: A297141 A004779 A247061 * A356450 A328189 A101766

Adjacent sequences: A180857 A180858 A180859 * A180861 A180862 A180863

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Sep 27 2010

STATUS

approved