Paths in graphs
We prove that if 10 ≦ (k2) ≦ m < (k+12) then the number of paths of length three in a graph G of size m is at most 2m(m - k)(k - 2)/k. Equality is attained iff G is the union of Kk and isolated vertices. We also give asymptotically best possible bounds for the maximal number of paths of length s, for arbitrary s, in graphs of size m. Lastly, we discuss the more general problem of maximizing the number of subgraphs isomorphic to a given graph H in graphs of size m.
Studia Scientiarum Mathematicarum Hungarica
Bollobás, B., & Sarkar, A. (2002). Paths in graphs. Studia Scientiarum Mathematicarum Hungarica, 38 (1-4), 115-137. https://doi.org/10.1556/sscmath.38.2001.1-4.8