Upper bound for 3-cycles in graphs of minimal degree 4.

Question

If we have a graph G of minimal degree 4 with v vertices and e edges, what is a good upper bound on the number of 3-cycles in G?

Answer 1

Absolutely, the trivial upper bound for 3-cycles is $\binom{n}{3}$ ($n$ is number of nodes). You should set a constraint on the maximum degree of the graph to get a better result.

Answer 2

There is an exact formula for the number $c_3$ of 3-cycles $$ c_3=\frac{1}{6}tr(A^3), $$ where $A$ is the adjacency matrix of graph.