# Number of 4 cycles

Let $$C_4$$ be a cycle with four 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.