 
Summary: ISRAELJOURNALOFMATHEMATICS,Vol.38,Nos.12,1981
ON THE NUMBER OF SUBGRAPHS
OF PRESCRIBED TYPE OF GRAPHS
WITH A GIVEN NUMBER OF EDGES*
BY
NOGAALON
ABSTRACT
All graphs considered are finite, undirected, with no loops, no multiple edges
and no isolated vertices. For a graph H=(V(H),E(H)) and for S C V(H)
define N(S) ={x ~ V(H):xy E E(H) for some y E S}. Define also ~(H) =
max{I S IIN(S)I:S C V(H)}, ,/(H) = ~,(IV(H)I + 8(H)). For two graphs G, H
let N(G, H) denote the number of subgraphs of G isomorphic to/4. Define also
for I > 0, N(L H) = max N(G, H), where the maximum is taken over all graphs
G with l edges. We investigate the asymptotic behaviour of N(l, H) for fixed H
as I tends to infinity. The main results are:
THEOREM A. ForeoerygraphH therearepositioeconstantsc~,c2such that
ctl~m~<=N(I,H)<=c,_l~'m foralll>=lE(H)l.
THEOREM B. If ~(H) = 0 then
1
Otl,2~.. (21)ml"~l/2N(I,H)=(I+ ., ,, ]AutHi
