A counterexample to the Fang--Lin conjecture: Edge and spectral extremality diverge near a Tur\'an graph

Abstract

Fang and Lin [J. Algebraic Combin. 63 (2026), Art.~58] asked whether, whenever $F$ is edge-color-critical with $\chi(F)=r+1$, every non-$r$-partite, $F$-free graph of maximum adjacency spectral radius must also maximize the number of edges. We give a negative answer. Let $F=K_1\vee\mu(K_3)$, where $\mu(K_3)$ is the Mycielskian of a triangle. This graph is edge-color-critical with $\chi(F)=5$. We prove that $\SPEX_{5}(n,F)\cap\EX_{5}(n,F)=\varnothing$ for all sufficiently large $n$. Thus no graph can simultaneously maximize both the edge count and the spectral radius.