Abstract
All graphs considered are finite, undirected, with no loops, no multiple edges and no isolated vertices. For a graphH=〈V(H),E(H)〉 and forS ⊂V(H) defineN(S)={x ∈V(H):xy ∈E(H) for somey ∈S}. Define alsoδ(H)= max {|S| − |N(S)|:S ⊂V(H)},γ(H)=1/2(|V(H)|+δ(H)). For two graphsG, H letN(G, H) denote the number of subgraphs ofG isomorphic toH. Define also forl>0,N(l, H)=maxN(G, H), where the maximum is taken over all graphsG withl edges. We investigate the asymptotic behaviour ofN(l, H) for fixedH asl tends to infinity. The main results are:Theorem A. For every graph H there are positive constants c 1, c2 such that {fx116-1}.
Theorem B. If δ(H)=0then {fx116-2},where |AutH|is the number of automorphisms of H.
(It turns out thatδ(H)=0 iffH has a spanning subgraph which is a disjoint union of cycles and isolated edges.)
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This paper forms part of an M.Sc. Thesis written by the author under the supervision of Prof. M. A. Perles from the Hebrew University of Jerusalem.
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Alon, N. On the number of subgraphs of prescribed type of graphs with a given number of edges. Israel J. Math. 38, 116–130 (1981). https://doi.org/10.1007/BF02761855
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DOI: https://doi.org/10.1007/BF02761855