A352666 Maximum number of induced copies of the claw graph K_{1,3} in an n-node graph.

0, 0, 0, 1, 4, 10, 20, 40, 70, 112, 176, 261, 372, 520, 704, 935, 1220, 1560, 1976, 2464, 3038, 3710, 4480, 5376, 6392, 7548, 8856, 10320, 11970, 13800, 15840, 18095, 20580, 23320, 26312, 29601, 33176, 37072, 41300, 45875, 50830, 56160, 61920, 68096, 74732

Offset: 1

Author

Pontus von Brömssen, Mar 26 2022

Keywords

• nonn

Comments

The sequence (a(n)/binomial(n,4)) is decreasing for n >= 4 and converges to 1/2, the inducibility of the claw graph.

Brown and Sidorenko (1994) prove that a bipartite optimal graph (i.e., an n-node graph with a(n) induced claw graphs) exists for all n. For n >= 2, the size k of the smallest part of an optimal bipartite graph K_{k,n-k} is one of the two integers closest to n/2 - sqrt(3*n/4-1), and a(n) = binomial(k,3)*(n-k) + binomial(n-k,3)*k. Both are optimal if and only if n is in A271713. For 7 <= n <= 10 (and, trivially, n = 3), the tripartite graph K_{1,1,n-2} is also optimal.

Links

• Jason I. Brown and Alexander Sidorenko, The inducibility of complete bipartite graphs, Journal of Graph Theory 18 (1994), 629-645.

Crossrefs

Cf. A271713.

Maximum number of induced copies of other graphs: A028723 (4-node cycle), A111384 (3-node path), A352665 (4-node path), A352667 (paw graph), A352668 (diamond graph), A352669 (cycles).

Programs

Python

from math import comb,isqrt
def A352666(n):
    if n <= 1: return 0
    r = isqrt(3*n-4)
    k0 = (n-r-1)//2
    return max(comb(k,3)*(n-k)+comb(n-k,3)*k for k in (k0,k0+1))