Find graphs on 5 vertices with 22 cycles, 13 cycles and 12 cycles

Question

This task is simple, but is there a solution based on some theory? Not just going through all graphs on 5 vertices.

Answer 1

One hint: any graph $G$ has a cycle space. Every element of this is a subset $S$ of $G$'s edges where each node of $G$ meets an even number of edges in $S$. Thus every cycle in $G$ is a member of the cycle space. Where $G$ has $n$ nodes, $m$ edges and $c$ components, $G$'s cycle space has $2^{m-n+c}$ elements. So if $n=5$ and $G$ has 22 cycles, $G$'s cycle space needs at least $2^5$ elements...