奇轮图上的Brualdi–Hoffman–Turán问题的密核方法
Dense-core approach to the Brualdi--Hoffman--Turán problem on odd wheels
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中文总结 AI 辅助
该研究采用密核方法，解决奇轮图的Brualdi–Hoffman–Turán问题，证明了不含$W_5$和$W_{2k+1}$的图的谱极值不等式，解决了Yu等人的猜想并强化了相关图类的已知结果。
中文摘要 AI 辅助
我们针对奇轮图$W_{2k+1}$（其中$k\geq2$，且$W_{2k+1}=K_1\vee C_{2k}$）的固定大小邻接谱极值问题，给出了统一表述。特殊情况$W_5$与一般情况$W_{2k+1}$（$k\ge3$）共享相同的密核归约和边谱稳定性，但具有不同的刚性结构。我们证明，每个足够大规模$m$的不含$W_5$的图$G$满足$\rho(G)^2-\rho(G)\le m$，当且仅当$G$为$K_{n,n}$且每个部分嵌入一个完美匹配时取等，其中$n$为偶数且$m=n^2+n$。对于任意固定的$k\ge3$，每个足够大规模$m$的不含$W_{2k+1}$的图$G$满足$\rho(G)^2-(k-1)\rho(G)\le m-\binom{k}{2}$，当且仅当$m=\binom{k}{2}+kq$时$G$为$K_k\vee qK_1$取等。我们的结果完全解决了Yu、Li和Peng提出的猜想，且通过一种不同的方法，进一步强化了关于足够大规模$m$的奇环、友谊图和奇扇图的已知结果。该证明结合了边谱稳定性定理、残差函数和密核方法。
英文摘要
We present a unified presentation of the fixed-size adjacency-spectral extremal problem for odd wheels $W_{2k+1}$, where $k\geq2$ and $W_{2k+1}=K_1\vee C_{2k}$. The exceptional case $W_5$ and the general case $W_{2k+1}$, $k\ge3$, share the same dense-core reduction and edge-spectral stability, but have different rigidity structures. We prove that every $W_5$-free graph of sufficiently large size $m$ satisfies $ρ(G)^2-ρ(G)\le m,$ with equality precisely for $K_{n,n}$ with a perfect matching embedded in each part, where $n$ is even and $m=n^2+n$. For any fixed $k\ge3$, every $W_{2k+1}$-free graph of sufficiently large size $m$ satisfies $ρ(G)^2-(k-1)ρ(G)\le m-\binom{k}{2},$ with equality precisely for $K_k\vee qK_1$ when $m=\binom{k}{2}+kq$. Our results completely settle a conjecture proposed by Yu, Li and Peng and, via a distinct approach, further strengthen known results concerning odd cycles, friendship graphs and odd fan graphs for sufficiently large $m.$ The proof combines the edge-spectral stability theorem, residual functions and the dense-core method.
