Spectral extremal graphs for $F_6$-free graphs with even size
Loujun Yu, Yuejian Peng
TL;DR
This work resolves the even-size extremal problem for the fan graph $F_6$ in the Brualdi-Hoffman-Turán setting. Using a spectral extremal approach based on the Perron vector, the authors partition the extremal graph around a Perron vertex and perform a delicate component-wise analysis to rule out all non-conforming structures, ultimately showing that for even $m\ge 3000$ the extremal graph is $S_{(m+4)/2,2}^{-1}$ and that equality in the spectral bound is achieved only by this graph. The result completes the picture for $F_6$-free graphs with even size and extends the previously known odd-size extremal characterization. The methods blend eigenvalue techniques with detailed structural constraints (via $U$, $W$ decompositions and $\eta(H),\gamma(w)$ metrics) to establish a unique extremal configuration with practical implications for spectral extremal graph theory of fan graphs.
Abstract
Let $F_l$ be the fan graph obtained by joining a vertex with a path on $l-1$ vertices. Yu, Li and Peng [Discrete Math. 346 (2023)] conjectured that if the number of edges of $G$ is $m$ and the spectral radius $λ(G)>\frac{k-1+\sqrt{4m-k^2+1}}{2}$, then $G$ contains a $F_{2k+1}$ and $F_{2k+2}$, unless $G=K_{k}\vee (\frac{m}{k}-\frac{k-1}{2})K_1$. The case $k\geq 3$ of the above conjecture has been confirmed by Li, Zhao and Zou [J. Graph theory 110 (2025)]. Zhang and Wang [Discrete Math. 347 (2024)], Yu, Li and Peng [Discrete Math. 348 (2025)], Gao and Li [Discrete Math. 349 (2026)] confirmed the case $k=2$. However, the extremal graphs for the case $k=2$ only exist when $m$ is odd. The case with $m$ even has not been determined. In this paper, we characterize the extremal graph for $F_6$ and even $m\ge 3000$.