Let $F$ be a connected color-critical graph with $\chi(F)=r+1\ge4$, let $S_{n,\Delta}^{(r)}=(n-\Delta)K_1\vee T(\Delta,r-1)$. We determine the graph of maximum adjacency spectral radius among all $n$-vertex $F$-free graphs with prescribed maximum degree $\Delta$. There is a constant $s_F\in[0,1)$ such that, for all sufficiently large $n$, $\left\lceil\frac{(r-1)n}{r}\right\rceil\le \Delta\le n-\Theta(n^{s_F})$ implies that every $n$-vertex $F$-free graph $G$ with $\Delta(G)=\Delta$ satisfies $\rho(G)\le \rho\bigl(S_{n,\Delta}^{(r)}\bigr)$, with equality if and only if $G\cong S_{n,\Delta}^{(r)}$. This is the spectral counterpart of the edge theorem of [European J. Combin. 106 (2022), 103576.] and extends the clique result in [arXiv:2608.26634, 2026.]. This result also provides a benchmark for unbalanced spectral Tur\'an problems arising from other extremal parameters.
Classical Tur\'an theory shows that, without additional constraints, the extremal structure of $F$-free graphs is governed by the chromatic number of $F$ and is asymptotically the balanced Tur\'an graph. We investigate how prescribing a large maximum degree changes this picture and leads to an unbalanced Tur\'an proble...
Let $Y_{n,r,q}$ be the graph obtained from the Tur\'an graph $T_{n,r}$ by adding $q$ pairwise disjoint edges inside a largest part, and let $c(n,F)$ be the minimum number of copies of $F$ created by adding a single edge to $T_{n,r}$. Fang, Li, Lin and Ma proved that for every color-critical graph $F$ with $\chi (F)=r+1...
We establish an entropy bridge, and then use it to prove that for any color-critical graph $F$ with chromatic number $\chi(F)=r+1\ge 3$, there exists a constant $\lambda_0=\lambda_0(F)$ such that if $G$ is an $F$-free graph with $\lambda (G)\ge \lambda_0$, then for every $\ell\ge 1$ with $(r,\ell )\neq (2,2)$, \[ \lamb...
Let $G$ be an $n$-vertex connected nonregular graph with maximum degree $\Delta$ that attains the maximum spectral radius. For $3\le\Delta\le n-2$, Liu and Li (2008) conjectured that $G$ has the degree sequence $(\Delta,\ldots,\Delta,\delta)$ with $\delta<\Delta$ as large as possible. In this paper, we study the dense...
The codegree Tur\'an density $\pi_{\mathrm{co}}(F)$ is the supremum over all $\gamma \in [0,1)$ such that, for arbitrarily large $n$, there exists an $n$-vertex $F$-free $k$-graph $H$ whose every $(k-1)$-subset of vertices lies in at least $\gamma n$ edges. Ding, Lamaison, Liu, Wang, and Yang (JLMS, 2025) studied the p...
For a connected graph $G$, let $\tau(G)$ denote the maximum number of pairwise edge-disjoint spanning trees, and let $\rho(G)$ be its adjacency spectral radius. For integers $n\ge1$, $k\ge2$, and $k\le\delta\le2k-1$, let $\mathcal{G}_{n,\delta}$ be the class of connected $n$-vertex graphs with minimum degree $\delta$,...
Xuan-Yu Cao, Chun-Xiang Wang· 0 citations
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