Dense-core approach to the Brualdi--Hoffman--Turán problem on odd wheels

Abstract

We present a unified presentation of the fixed-size adjacency-spectral extremal problem for odd wheels $W_{2k+1}$, where $k\ge 2$ and $W_{2k+1}=K_1\vee C_{2k}$. The exceptional case $W_5$ and the general case $W_{2k+1}$, $k\ge 3$, 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 $\rho (G{)}^2-\rho (G)\le m,$ 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 $k\ge 3$, every $W_{2k+1}$-free graph of sufficiently large size $m$ satisfies $\rho (G{)}^2-(k-1)\rho (G)\le m-(\genfrac{}{}{0ex}{}{k}{2}),$ with equality precisely for $K_k\vee q{K}_1$ when $m=\left(\genfrac{}{}{0ex}{}{k}{2}\right)+kq$. 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.