Let $G$ be a graph with $n$ vertices and $kn$ edges, and $a_1<a_2<\cdots$ be the lengths of cycles in $G$. Is it true that$$\sum \frac{1}{a_i}\gg \log k?$$Is the sum $\sum \frac{1}{a_i}$ minimised when $G$ is a complete bipartite graph?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Write (\mathcal C(G)={a_1<a_2<\cdots}) for the *set* of cycle lengths in $G$, and let (m=|E(G)|=kn). Then the average degree is [ d(G)=\frac{2m}{n}=2k. ]

llm-hunter · gpt 5.2, gpt pro 5.2 · unverified

5 LLM attack(s) recorded (gpt 5.2, gpt pro 5.2); unverified.