Multiplicities of sums in the explicit formulae for counting fixed length cycles in undirected graphs

An explicit formula for counting $k$-cycles in graphs is the combination of sums corresponding to the shapes of closed $k$-walks. It was shown that the maximum multiplicity of a sum in the formula is $[k/2]$ for, starting with $k=8$. In this work, we study the maximum sum multiplicity for some families of graphs: bipartite, triangle-free, planar, maximum vertex degree three, and their intersections. When $k$ is large, the biparticity and degree boundednesses are the only properties which decrease the maximum sum multiplicity by 1, providing $k\equiv2,3\pmod4$. Some combinations of properties in the case of $k\leq20$ yield the decrease by 1 or 2.