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number of 4 cycles
Let $C_4$ be a cycle with $4$ vertices. For an arbitrary graph $G$ with $n$ vertices and m edges say $m>n\sqrt n$, how many $C_4$s exist? Is there a lower bound for this?
Yes, this is known. For $d = \Omega(n^{1/2})$ with a sufficiently large implicit constant, any $n$node graph of average degree $d$ has $\Omega(d^4)$ total $C_4$s.
This is best possible because it's realized by a random graph.
The earliest reference I'm aware of for this is "CubeSupersaturated Graphs and Related Problems" by Erdos and Simonovits, where it's claimed without proof. There are many proofs out there, off the top of my head see Lemma 3 here.

Yes, this is known. For $d = \Omega(n^{1/2})$ with a sufficiently large implicit constant, any $n$node graph of average degree $d$ has $\Omega(d^4)$ total $C_4$s.
This is best possible because it's realized by a random graph.
The earliest reference I'm aware of for this is "CubeSupersaturated Graphs and Related Problems" by Erdos and Simonovits, where it's claimed without proof. There are many proofs out there, off the top of my head see Lemma 3 here.
20181015 15:34:34