October 03, 2025
Let \(Q\) be an \(s\)-vertex \(r\)-uniform hypergraph, and let \(H\) be an \(n\)-vertex \(r\)-uniform hypergraph. Denote by \(\mathcal{N}(Q,H)\) the number of isomorphic copies of \(Q\) in \(H\). For a hereditary family \(\mathcal{P}\) of \(r\)-uniform hypergraphs, define \[\pi(Q,\mathcal{P}):=\lim\limits_{n\to \infty}\binom{n}{s}^{-1}\max\{\mathcal{N}(Q,H): H\in \mathcal{P}~~and~~|V(H)|=n\}.\] For \(p\geq1\), the \((p,Q)\)-spectral radius of \(H\) is defined as \[\lambda^{(p)}(Q,H)=\max_{\|\mathbf{x}\|_{p}=1}s!\sum_{\{i_{1},\ldots,i_{s}\}\in \binom{[n]}{s}}\mathcal{N}(Q,H[\{i_{1},\ldots,i_{s}\}])x_{i_{1}}\cdots x_{i_{s}}.\]
In this paper, we present a systematically investigation of the parameter \(\lambda^{(p)}(Q,H)\). First, we prove that the limit \[\lambda^{(p)}(Q,\mathcal{P}):=\lim\limits_{n\to \infty}n^{s/p-s}\max\{\lambda^{(p)}(Q,H): H\in \mathcal{P}~~and~~|V(H)|=n\}\] exists, and for \(p>1\), it satisfies \[\pi(Q,\mathcal{P})=\lambda^{(p)}(Q,\mathcal{P}).\] Second, we study spectral generalized Turán problems. Specifically, we establish a spectral stability result and apply it to derive a spectral version of the Erdős Pentagon Problem: for \(p\geq1\) and sufficiently large \(n\), the balanced blow-up of \(C_{5}\) maximizes \(\lambda^{(p)}(C_{5},H)\) among all \(n\)-vertex triangle-free graphs \(H\), thereby improving a result of Liu [1]. Furthermore, we show that for \(p\geq1\) and sufficiently large \(n\), the \(l\)-partite Turán graph \(T_{l}(n)\) attains the maximum \(\lambda^{(p)}(K_{s},H)\) among all \(n\)-vertex F-free graphs \(H\), where \(F\) is an edge-critical graph with \(\chi(F)=l+1\). This provides a spectral analogue of a theorem due to Ma and Qiu [2].
A hypergraph \(H=(V(H),E(H))\) consists of a vertex set \(V(H)=\{v_1,v_2,{\cdots},v_n\}\) and an edge set \(E(H)=\{e_1,e_2,{\cdots},e_m\}\) , where \(e_i \subseteq V\) for \(i \in [m]:=\{1,2,\ldots,m\}\). The order and size of \(H\) are defined as \(\nu(H):=|V(H)|\) and \(e(H):=|E(H)|\), respectively. If \(|e_i|=r\) for each \(i \in [m]\) and \(r \geq2\), then \(H\) is called an \(r\)-uniform hypergraph (or \(r\)-graph). A simple graph is exactly a \(2\)-uniform hypergraph. The subgraph \(H[I]\) is induced by \(I\), that is, \(V(H[I])=I\) and \(E(H[I])=\{e\in E(H):e\subseteq I\}\). For any vertex \(v\in V(H)\), we write \(H-v\) for the subgraph of \(H\) induced by \(V(H)\backslash\{v\}\). For \(l\geq r\geq2\), an \(r\)-graph is called \(l\)-partite if its vertex set can be divided into \(l\) parts such that each edge has at most one vertex from each part. An edge maximal \(l\)-partite \(r\)-graph is called complete \(l\)-partite. Let \(T^{r}_{l}(n)\) be the complete \(l\)-partite \(r\)-graph on \(n\) vertices without two part sizes differing by more than one; when \(r=2\), the graph \(T^{2}_{l}(n)\) is Turán graph \(T_{l}(n)\).
Given an \(s\)-vertex \(r\)-graph \(Q\) and an \(r\)-graph \(H\), let \(\mathcal{N}(Q,H)\) denote the number of isomorphic copies of \(Q\) in \(H\). For example, for the complete \(r\)-graph \(K_{s}^{r}\) on \(s\) vertices, we have \(\mathcal{N}(Q,K_{s}^{r})=\frac{s!}{|Aut(Q)|}\). For a family \(\mathcal{F}\) of \(r\)-graphs, we say a hypergraph \(G\) is \(\mathcal{F}\)-free if \(G\) does not contain any member of \(\mathcal{F}\) as a subgraph. The generalized Turán number \(ex(n,Q,\mathcal{F})\) is the largest \(\mathcal{N}(Q,H)\) among all the \(n\)-vertex \(\mathcal{F}\)-free \(r\)-graphs \(H\). The function \(ex(n,Q,\mathcal{F})\) is a well-studied parameter; a comprehensive survey can be found in [3]. Let \(E(Q,H)\) denote the collection of all \(s\)-subsets \(I\) of \(V(H)\) such that \(\mathcal{N}(Q,H[I])>0\), and define \(E_{Q,H}(v)=\{I\in E(Q,H):v\in I\}\). The \(Q\)-degree of \(v\), denoted \(d_{Q,H}(v)\), is given by \[d_{Q,H}(v)=\sum_{I\in E_{Q,H}(v)} \mathcal{N}(Q,H[I]).\] The minimum \(Q\)-degree of \(H\) is denoted by \(\delta_{Q}(H)\).
Let \(p\geq1\), \(Q\) be an \(s\)-vertex \(r\)-graph and \(H\) be an \(n\)-vertex \(r\)-graph, where \(r\leq s\leq n\). The Q-Lagrangian polynomial \(P_{Q,H}(\mathbf{x})\) of \(H\) is defined as \[\begin{align} P_{Q,H}(\mathbf{x}) &=s!\sum_{\{i_{1},\ldots,i_{s}\}\in \binom{[n]}{s}}\mathcal{N}(Q,H[\{i_{1},\ldots,i_{s}\}])x_{i_{1}}\cdots x_{i_{s}}\\ &=s!\sum_{\{i_{1},\ldots,i_{s}\}\in E(Q,H)}\mathcal{N}(Q,H[\{i_{1},\ldots,i_{s}\}])x_{i_{1}}\cdots x_{i_{s}}, \end{align}\] and the \((p,Q)\)-spectral radius \(\lambda^{(p)}(Q,H)\) of \(H\) is defined as \[\lambda^{(p)}(Q,H)=\max_{\|\mathbf{x}\|_{p}=1}P_{Q,H}(\mathbf{x}),\] where \(\mathbf{x}=(x_{1},\ldots,x_{n})\in \mathbb{R}^{n}\) and \(\|\mathbf{x}\|_{p}:=(|x_{1}|^{p}+\cdots +|x_{n}|^{p})^{1/p}\). It is noteworthy that the definition of the \((p,Q)\)-spectral radius was recently introduced by Liu [1], and our definition here differs from Liu’s by a constant factor \(\frac{|Aut(Q)|}{s!}\). When \(Q=K_{s}^{r}\), we abbreviate \(\lambda^{(p)}(Q,H)\) as \(\lambda^{(p)}_{s}(H)\), termed the \(s\)-clique \(p\)-spectral radius of \(H\). If, further \(Q=K_{r}^{r}\), we simple write \(\lambda^{(p)}(H)\), recovering the \(p\)-spectral radius of \(H\) introduced by Keevash, Lenz, and Mubayi[4]. If \(\mathbf{x}\in \mathbb{R}^{n}\) is a vector such that \(\|\mathbf{x}\|_{p}=1\) and \(\lambda^{(p)}(Q,H)=P_{Q,H}(\mathbf{x})\), then \(\mathbf{x}\) is called a \(Q\)-eigenvector of \(H\) corresponding to \(\lambda^{(p)}(Q,H)\). Clearly, there always exists a nonnegative \(Q\)-eigenvector corresponding to \(\lambda^{(p)}(Q,H)\), called a principal \(Q\)-eigenvector of \(H\). Moreover, if a principal \(Q\)-eigenvector \(\mathbf{x}\) is strictly positive (i.e., \(x_{v}>0\) for all \(v\in V(H)\)), then we call it a Perron-Frobenius \(Q\)-eigenvector of \(H\).
A property of \(r\)-graphs is a family of \(r\)-graphs closed under isomorphisms. For a property \(\mathcal{P}\), denoted by \(\mathcal{P}_{n}\) the collection of \(r\)-graphs in \(\mathcal{P}\) of order \(n\). A property is called hereditary if it is closed under taking induced subgraphs. Given a family \(\mathcal{F}\) of \(r\)-graphs, the class of all \(\mathcal{F}\)-free \(r\)-graphs forms a hereditary property, denoted by \(\overline{\mathcal{F}}\). Throughout our discussion, we always assume that the hereditary property of \(r\)-graphs is closed under taking the disjoint union with an isolated vertex. Given two \(r\)-graphs \(Q\) and \(H\), a map \(\phi\): \(V(Q)\rightarrow V(H)\) is a homomorphism from \(Q\) to \(H\) if \(\phi(e)\in E(H)\) for all \(e\in E(Q)\). We say \(Q\) is \(H\)-colorable if there is a homomorphism from \(Q\) to \(H\).
A fundamental problem in extremal combinatorics can be formulated as follows: Given an \(s\)-vertex \(r\)-graph \(Q\) and a hereditary property \(\mathcal{P}\) of \(r\)-graphs, determine the extremal function \[ex(Q,\mathcal{P}_{n}):=\max_{H\in \mathcal{P}_{n}}\mathcal{N}(Q,H).\] By Katona-Nemetz-Simonovits averaging argument [5], the ratio \(ex(Q,\mathcal{P}_{n})/\binom{n}{s}\) is decreasing in \(n\), and so the limit
\[\label{eltt} \pi(Q,\mathcal{P}):=\lim\limits_{n\to \infty}\frac{ex(Q,\mathcal{P}_{n})}{\binom{n}{s}}\tag{1}\] always exists, called the \(Q\)-density of \(\mathcal{P}\). If \(\mathcal{P}=\overline{\mathcal{F}}\) for a family \(\mathcal{F}\) of \(r\)-graphs, then \(ex(K_{r}^{r},\mathcal{P}_{n})\) and \(\pi(K_{r}^{r},\mathcal{P})\) are the Turán number and Turán density of \(\mathcal{F}\), respectively. To maintain consistency in notation, we will use \(ex(Q,\overline{\mathcal{F}}_{n})\) instead of \(ex(n,Q,\mathcal{F})\) in the remaining part.
Similarly, we can study the spectral analogue of the aforementioned problem. For an \(s\)-vertex \(r\)-graph \(Q\) and a hereditary property \(\mathcal{P}\) of \(r\)-graphs, we define \[\lambda^{(p)}(Q,\mathcal{P}_{n}):=\max_{H\in \mathcal{P}_{n}}\lambda^{(p)}(Q,H),\] and the \((p,Q)\)-spectral density of \(\mathcal{P}\) is defined as \[\lambda^{(p)}(Q,\mathcal{P}):=\lim\limits_{n\to \infty}\frac{\lambda^{(p)}(Q,\mathcal{P}_{n})}{n^{s-s/p}}.\]
In [6], Nikiforov conducted a systematic study of the \(p\)-spectral radius of hypergraphs using analytical methods, and proved that \(\pi(K_{r}^{r},\mathcal{P})=\lambda^{(p)}(K_{r}^{r},\mathcal{P})\) holds for any \(p>1\) and any hereditary property \(\mathcal{P}\) of \(r\)-graphs. Liu and Bu [7] introduced the \(s\)-clique spectral radius of a graph \(G\)(equivalent to \(\lambda^{(s)}_{s}(G)\)), and extended the spectral Mantel’s theorem via the clique tensor. Yu and Peng [8] gave a spectral version of the generalized Erdős-Gallai theorem via the clique tensor. In [1], Liu established a general theorem that extends the result of Keevash-Lenz-Mubayi and obtained a spectral Erdős pentagon theorem.
In this paper, we investigate spectral extremal problems concerning the \((p,Q)\)-spectral radius of hypergraphs. For any hereditary property \(\mathcal{P}\) of \(r\)-graphs, we prove that the \((p,Q)\)-spectral density of \(\mathcal{P}\) exists for all \(p\geq1\). Moreover, we show that the \(Q\)-density of \(\mathcal{P}\) coincides with its \((p,Q)\)-spectral density when \(p>1\). Furthermore, we study spectral generalized Turán problems. In particular, we establish a spectral stability result: if the maximum \((p,Q)\)-spectral radius among all \(\mathcal{F}\)-free \(r\)-graphs satisfies a specific growth condition, then the extremal hypergraphs must possess a large minimum \(Q\)-degree. As an application, we derive a spectral analogue of the Erdős Pentagon Problem. Specifically, for any \(p\geq1\) and all sufficiently large \(n\), the balanced blow-up of \(C_{5}\) attains the maximal \((p,C_{5})\)-spectral radius over all \(n\)-vertex triangle-free graphs. This extends the result of Liu [1]. Additionally, we demonstrate that for \(p\geq1\) and \(n\) sufficiently large, the \(l\)-partite Turán graph \(T_{l}(n)\) achieves the maximum \(s\)-clique \(p\)-spectral radius among all \(n\)-vertex \(F\)-free graphs, where \(F\) is an edge-critical graph with \(\chi(F)=l+1\). This establishes a spectral counterpart to the result of Ma and Qiu [2] and extends a theorem of Yu and Peng [8].
In this section, we present some fundamental properties of the parameter \(\lambda^{(p)}(Q,H)\). Hereafter, when given an \(s\)-vertex \(r\)-graph \(Q\) and an \(n\)-vertex \(r\)-graph \(H\), it is always assumed that \(n\geq s\geq r\geq 2\), provided no ambiguity arises.
Proposition 1. Let \(Q\) be an \(s\)-vertex \(r\)-graph and \(H\) be an \(n\)-vertex \(r\)-graph. If \(p\geq1\), then \(\lambda^{(p)}(Q,H)\) is an increasing and continuous function in \(p\). Moreover, \[\lim\limits_{p\to \infty}\lambda^{(p)}(Q,H)=s!\mathcal{N}(Q,H).\]
Proof. Since \(\lambda^{(p)}(Q,H)\) always has a nonnegative \(Q\)-eigenvector, we obtain the following equivalent definition of \(\lambda^{(p)}(Q,H)\): \[\label{edef} \lambda^{(p)}(Q,H)=\max_{|x_{1}|+\cdots+|x_{n}|=1}s!\sum_{\{i_{1},\ldots,i_{s}\}\in E(Q,H)}\mathcal{N}(Q,H[\{i_{1},\ldots,i_{s}\}])|x_{i_{1}}|^{1/p}\cdots |x_{i_{s}}|^{1/p},\tag{2}\] where \(\mathbf{x}=(x_{1},\ldots,x_{n})\in \mathbb{R}^{n}\). Note that \(0\leq|x_{i_{1}}|\cdots |x_{i_{s}}|\leq1\). We now claim that for any \(b\geq a\geq1\), \[0\leq|x_{i_{1}}|^{1/b}\cdots |x_{i_{s}}|^{1/b}-|x_{i_{1}}|^{1/a}\cdots |x_{i_{s}}|^{1/a}\leq b-a.\] The left inequality holds trivially, and the right equality holds when \(|x_{i_{1}}|\cdots |x_{i_{s}}|=0 ~~or~~1\). For \(0<|x_{i_{1}}|\cdots |x_{i_{s}}|<1\), we apply the Mean Value Theorem to the function \(f(x):=(|x_{i_{1}}|\cdots |x_{i_{s}}|)^{1/x}\). There exists \(\xi\in (a,b)\) such that \[\begin{align} |x_{i_{1}}|^{1/b}\cdots |x_{i_{s}}|^{1/b}-|x_{i_{1}}|^{1/a}\cdots |x_{i_{s}}|^{1/a} &=(b-a)\xi^{-2}(|x_{i_{1}}|\cdots |x_{i_{s}}|)^{\xi^{-1}}\ln(|x_{i_{1}}|\cdots |x_{i_{s}}|)^{-1}\\ &\leq (b-a)(|x_{i_{1}}|\cdots |x_{i_{s}}|)^{\xi^{-1}-\xi^{-2}}\\ &\leq b-a. \end{align}\] Thus, the claim holds.
Let \(\mathbf{y}=(y_1,\ldots,y_n)\) be a nonnegative vector such that equality (2 ) holds for \(\lambda^{(a)}(Q,H)\). Then, \[\lambda^{(b)}(Q,H)-\lambda^{(a)}(Q,H)\geq s!\sum_{\{i_{1},\ldots,i_{s}\}\in E(Q,H)}\mathcal{N}(Q,H[\{i_{1},\ldots,i_{s}\}])(y_{i_{1}}^{1/b}\cdots y_{i_{s}}^{1/b}-y_{i_{1}}^{1/a}\cdots y_{i_{s}}^{1/a})\geq 0.\] This implies that \(\lambda^{(p)}(Q,H)\) is increasing in \(p\).
Now, let \(\mathbf{z}=(z_1,\ldots,z_n)\) be a nonnegative vector such that equality (2 ) holds for \(\lambda^{(b)}(Q,H)\). Then, \[\begin{align} 0\leq\lambda^{(b)}(Q,H)-\lambda^{(a)}(Q,H)&\leq s!\sum_{\{i_{1},\ldots,i_{s}\}\in E(Q,H)}\mathcal{N}(Q,H[\{i_{1},\ldots,i_{s}\}])(z_{i_{1}}^{1/b}\cdots z_{i_{s}}^{1/b}-z_{i_{1}}^{1/a}\cdots z_{i_{s}}^{1/a})\\ &\leq (b-a)s!\mathcal{N}(Q,H). \end{align}\] Therefore, \(\lambda^{(p)}(Q,H)\) satisfies the Lipschitz condition and is thus continuous.
By the definition of \(\lambda^{(p)}(Q,H)\), it is evident that \(\lambda^{(p)}(Q,H)\leq s!\mathcal{N}(Q,H)\). On the other hand, taking the \(n\)-vector \(\mathbf{x}=(n^{-1/p},\ldots,n^{-1/p})\) yields \[\lambda^{(p)}(Q,H)\geq P_{Q,H}(\mathbf{x})=s!\mathcal{N}(Q,H)/n^{s/p}.\] Thus, we obtain \[s!\mathcal{N}(Q,H)/n^{s/p}\leq \lambda^{(p)}(Q,H)\leq s!\mathcal{N}(Q,H),\] which implies \(\lim\limits_{p\to \infty}\lambda^{(p)}(Q,H)=s!\mathcal{N}(Q,H)\). This completes the proof. ◻
For a vertex subset \(U\subseteq V(H)\) of an \(n\)-vertex \(r\)-graph \(H\), we write \(x_{U}=\Pi_{v\in U}x_{v}\). For \(p>1\), the principal Q-eigenvector \(\mathbf{x}=(x_{1},\ldots,x_{n})\) of \(H\) satisfies the following system eigenequations derived from Lagrange’s method: \[\label{eigen} \lambda^{(p)}(Q,H)x_{i}^{p-1}=(s-1)!\sum_{I\in E_{Q,H}(i)}\mathcal{N}(Q,H[I])x_{I\backslash\{i\}},~ for~ 1\leq i\leq n.\tag{3}\]
Lemma 2. Let \(p\geq1\), and let \(Q\) be an \(s\)-vertex \(r\)-graph and \(H\) be an \(n\)-vertex \(r\)-graph. Then function \[f_{Q,H}(p)=\bigg(\frac{\lambda^{(p)}(Q,H)}{s!\mathcal{N}(Q,H)}\bigg)^{p}\] is decreasing in \(p\).
Proof. Set \(\beta\geq\alpha\geq1\) and \(\mathcal{N}:=\mathcal{N}(Q,H)\). Let \(\mathbf{x}=(x_{1},\ldots,x_{n})\) be a principal \(Q\)-eigenvector corresponding to \(\lambda^{(\beta)}(Q,H)\). Using Power-Mean inequality, we obtain \[\frac{\lambda^{(\beta)}(Q,H)}{s!\mathcal{N}}=\frac{1}{\mathcal{N}} \sum_{I\in E(Q,H)}\mathcal{N}(Q,H[I])x_{I} \leq\bigg(\frac{1}{\mathcal{N}} \sum_{I\in E(Q,H)}\mathcal{N}(Q,H[I])(x_{I})^{\beta/\alpha}\bigg)^{\alpha/\beta}.\] Note that \[\big(x_{1}^{\beta/\alpha}\big)^{\alpha}+\cdots+\big(x_{n}^{\beta/\alpha}\big)^{\alpha}=x_{1}^{\beta}+\cdots+x_{n}^{\beta}=1.\] Thus, we have \[\frac{1}{\mathcal{N}} \sum_{I\in E(Q,H)}\mathcal{N}(Q,H[I])(x_{I})^{\beta/\alpha}\leq\frac{1}{s!\mathcal{N}} \lambda^{(\alpha)}(Q,H),\] and so \[\bigg(\frac{\lambda^{(\beta)}(Q,H)}{s!\mathcal{N}}\bigg)^{\beta}\leq \bigg(\frac{\lambda^{(\alpha)}(Q,H)}{s!\mathcal{N}}\bigg)^{\alpha},\] completing the proof. ◻
We conclude this section with the following obvious result.
Proposition 3. Let \(p\geq1\), and let \(Q\) be an \(s\)-vertex \(r\)-graph and \(H\) be an \(n\)-vertex \(r\)-graph. If \(G\) is a subgraph of \(H\), then \(\lambda^{(p)}(Q,G)\leq \lambda^{(p)}(Q,H).\)
In this section, we show that for any hereditary property \(\mathcal{P}\) of \(r\)-graphs, the \(Q\)-density of \(\mathcal{P}\) is equal to its \((p,Q)\)-spectral density when \(p>1\), namely \(\pi(Q,\mathcal{P})=\lambda^{(p)}(Q,\mathcal{P})\). We then investigate the \((p,Q)\)-spectral radius of hereditary families which satisfy \(\pi(Q,\mathcal{P})=\lambda^{(1)}(Q,\mathcal{P})\).
Fact 4 ([9]). If \(p>1\) and \(s\geq2\), then the function \[f(x)=\frac{1-sx}{(1-x)^{s/p}}\] is decreasing for \(0\leq x<1\).
For a vector \(\mathbf{x}\in \mathbb{R}^{n}\), we use the notation \(\mathbf{x}_{\mathrm{min}}\) to represent the smallest element in the vector \(\mathbf{x}\).
Theorem 5. Let \(p\geq 1\), and let \(Q\) be an \(r\)-graph on \(s\) vertices. If \(\mathcal{P}\) is a hereditary property of \(r\)-graphs, then the limit \[\lambda^{(p)}(Q,\mathcal{P})=\lim\limits_{n\to \infty}\lambda^{(p)}(Q,\mathcal{P}_{n})n^{s/p-s}\] exists. If \(p=1\), then \(\lambda^{(1)}(Q,\mathcal{P}_{n})\) is increasing, and so \[\lambda^{(1)}(Q,\mathcal{P}_{n})\leq \lambda^{(1)}(Q,\mathcal{P}).\] If \(p>1\), then \(\lambda^{(p)}(Q,\mathcal{P})\) satisfies \[\lambda^{(p)}(Q,\mathcal{P})\leq\frac{\lambda^{(p)}(Q,\mathcal{P}_{n})n^{s/p}}{(n)_{s}},\] where \((n)_{s}=n(n-1)\cdots (n-s+1)\).
Proof. Let \(H\in \mathcal{P}_{n}\) be an \(r\)-graph satisfying \(\lambda^{(p)}(Q,H)=\lambda^{(p)}(Q,\mathcal{P}_{n})\), and let \(\mathbf{x}=(x_1,\ldots,x_n)\) be a principal \(Q\)-eigenvector corresponding to \(\lambda^{(p)}(Q,H)\). Through previous assumption about hereditary properties, we have \[\lambda^{(p)}(Q,\mathcal{P}_{n})\leq \lambda^{(p)}(Q,H+u)\leq \lambda^{(p)}(Q,\mathcal{P}_{n+1}),\] where \(u\notin V(H)\) and \(H+u\in \mathcal{P}_{n+1}\) is an \(r\)-graph with vertex set \(V(H+u)=V(H)\cup \{u\}\) and edge set \(E(H+u)=E(H)\). Thus, \(\lambda^{(p)}(Q,\mathcal{P}_{n})\) is increasing in \(n\).
Recall that \(\mathcal{N}(Q,K_{s}^{r})=\frac{s!}{|Aut(Q)|}\leq s!\). For \(p=1\), by Maclaurin’s inequality, we have \[\lambda^{(1)}(Q,\mathcal{P}_{n})\leq s!\sum_{\{i_{1},\ldots,i_{s}\}\in \binom{[n]}{s}}s!x_{i_{1}}\cdots x_{i_{s}}\leq s!(x_{1}+\cdots+x_{n})^{s}= s!.\] Thus, the sequence \(\Big\{\lambda^{(1)}(Q,\mathcal{P}_{n})\Big\}_{n=1}^{\infty}\) converges to a limit \(\lambda\), and we conclude \[\lambda=\lim\limits_{n\to \infty}\lambda^{(1)}(Q,\mathcal{P}_{n})n^{s-s}=\lambda^{(1)}(Q,\mathcal{P}).\] For \(p>1\), let \(k\in V(H)\) be a vertex with \(x_{k}=\mathbf{x}_{\mathrm{min}}\), and let \(\mathbf{x}'\) be the \((n-1)\)-vector obtained from \(\mathbf{x}\) by removing the component \(x_{k}\). By (3 ), we have \[P_{Q,H-k}(\mathbf{x}')=\lambda^{(p)}(Q,H)-s!x_{k}\sum_{I\in E_{Q,H}(k)}\mathcal{N}(Q,H[I])x_{I\backslash\{k\}}= \lambda^{(p)}(Q,\mathcal{P}_{n})-s\lambda^{(p)}(Q,\mathcal{P}_{n})x_{k}^{p}.\] Since \(\mathcal{P}\) is hereditary, \(H-k\in \mathcal{P}_{n-1}\). Therefore, \[\lambda^{(p)}(Q,\mathcal{P}_{n})(1-sx_{k}^{p})=P_{Q,H-k}(\mathbf{x}')\leq \lambda^{(p)}(Q,H-k)(\|\mathbf{x}' \|_{p}^{s}) \leq \lambda^{(p)}(Q,\mathcal{P}_{n-1})(1-x_{k}^{p})^{s/p},\] or equivalently, \[\label{e8} \frac{\lambda^{(p)}(Q,\mathcal{P}_{n-1})}{\lambda^{(p)}(Q,\mathcal{P}_{n})}\geq \frac{1-sx_{k}^{p}}{(1-x_{k}^{p})^{s/p}}.\tag{4}\] Note that \((\mathbf{x}_{\mathrm{min}})^{p}\leq 1/n\), by (4 ) and Fact 4, we have \[\frac{\lambda^{(p)}(Q,\mathcal{P}_{n-1})}{\lambda^{(p)}(Q,\mathcal{P}_{n})} \geq\frac{1-s(\mathbf{x}_{\mathrm{min}})^{p}}{(1-(\mathbf{x}_{\mathrm{min}})^{p})^{s/p}} \geq\frac{1-s/n}{(1-1/n)^{s/p}}.\] This implies that \[\frac{\lambda^{(p)}(Q,\mathcal{P}_{n-1})(n-1)^{s/p}}{(n-1)_{s}}\geq \frac{\lambda^{(p)}(Q,\mathcal{P}_{n})n^{s/p}}{(n)_{s}}.\] Therefore, the sequence \(\Big\{\frac{\lambda^{(p)}(Q,\mathcal{P}_{n})n^{s/p}}{(n)_{s}}\Big\}_{n=1}^{\infty}\) is decreasing and hence convergent. This completes the proof. ◻
Given a hereditary property \(\mathcal{P}\) of \(r\)-graphs. For \(H\in \mathcal{P}_{n}\) with \(\mathcal{N}(Q,H)=ex(Q,\mathcal{P}_{n})\), the \(n\)-vector \(\mathbf{x}=(n^{-1/p},\ldots,n^{-1/p})\) yields \[\label{zhy} \lambda^{(p)}(Q,H)\geq P_{Q,H}(\mathbf{x})=s!\mathcal{N}(Q,H)/n^{s/p} =s!ex(Q,\mathcal{P}_{n})/n^{s/p}.\tag{5}\] Thus \[\lambda^{(p)}(Q,\mathcal{P}_{n})\geq \lambda^{(p)}(Q,H) \geq s!ex(Q,\mathcal{P}_{n})/n^{s/p},\] which implies \[\frac{\lambda^{(p)}(Q,\mathcal{P}_{n})n^{s/p}}{(n)_{s}}\geq \frac{ex(Q,\mathcal{P}_{n})}{\binom{n}{s}}.\] Taking \(n\to \infty\) and applying Theorem 5, we obtain for \(p\geq1\), \[\label{pp} \lambda^{(p)}(Q,\mathcal{P})\geq\pi(Q,\mathcal{P}).\tag{6}\]
We now state one of our main results: we show that for \(p>1\), equality in inequality (6 ) always holds. This significantly extends the result of Nikiforov [6].
Theorem 6. If \(Q\) is an \(r\)-graph and \(\mathcal{P}\) is a hereditary property of \(r\)-graphs, then for every \(p>1\), \[\lambda^{(p)}(Q,\mathcal{P})=\pi(Q,\mathcal{P}).\]
In the following, we present several lemmas necessary for the proof of Theorem 6.
Lemma 7. Let \(Q\) be an \(r\)-graph on \(s\) vertices, and \(\mathcal{P}\) be a hereditary property of \(r\)-graphs with \(\lambda^{(p)}(Q,\mathcal{P})>0\). If \(p>1\) and \(\lambda_{n}^{(p)}:= \lambda^{(p)}(Q,\mathcal{P}_{n})\), then there exist infinitely many \(n\) such that \[\frac{\lambda_{n-1}^{(p)}(n-1)^{s/p}}{(n-1)_{s}}-\frac{\lambda_{n}^{(p)}n^{s/p}}{(n)_{s}}<\frac{1}{n\log n}\cdot\frac{\lambda_{n}^{(p)}n^{s/p}}{(n)_{s}}.\]
Proof. Assume for a contradiction that there exists \(n_{0}\) such that for all \(n\geq n_{0}\), \[\frac{\lambda_{n-1}^{(p)}(n-1)^{s/p}}{(n-1)_{s}}-\frac{\lambda_{n}^{(p)}n^{s/p}}{(n)_{s}}\geq\frac{1}{n\log n}\cdot\frac{\lambda_{n}^{(p)}n^{s/p}}{(n)_{s}}.\] Summing the inequalities for all \(n_0,n_0+1,\ldots, k\), we get \[\begin{align} \frac{\lambda_{n_{0}-1}^{(p)}(n_{0}-1)^{s/p}}{(n_{0}-1)_{s}}-\frac{\lambda_{k}^{(p)}k^{s/p}}{(k)_{s}} &=& \sum_{n=n_{0}}^{k} \left( \frac{\lambda_{n-1}^{(p)}(n-1)^{s/p}}{(n-1)_{s}}-\frac{\lambda_{n}^{(p)}n^{s/p}}{(n)_{s}} \right) \\ &\geq& \sum_{n=n_{0}}^{k} \frac{1}{n\log n}\cdot\frac{\lambda_{n}^{(p)}n^{s/p}}{(n)_{s}}\\ &\geq& \lambda^{(p)}(Q,\mathcal{P})\sum_{n=n_{0}}^{k} \frac{1}{n\log n}, \end{align}\] where the last inequality follows from Theorem 5. Note that the left-hand side is bounded and the right-hand side diverges. Taking \(k\) sufficiently large, we obtain a contradiction. ◻
Lemma 8. Let \(Q\) be an \(r\)-graph on \(s\) vertices, and \(\mathcal{P}\) be a hereditary property of \(r\)-graphs with \(\lambda^{(p)}(Q,\mathcal{P})>0\). Suppose that \(H_{n}\in \mathcal{P}_{n}\) is an \(r\)-graph satisfying \(\lambda^{(p)}(Q,H_{n})=\lambda^{(p)}(Q,\mathcal{P}_{n})\) for \(p>1\) and \(\mathbf{x}=(x_1,\ldots,x_n)\) is a principal \(Q\)-eigenvector corresponding to \(\lambda^{(p)}(Q,H_{n})\). Then there exist infinitely many \(n\) such that \[(\mathbf{x}_{\mathrm{min}})^{p}\geq\frac{1}{n}\Big(1-\frac{p}{(p-1)s\log n}\Big).\]
Proof. Assume, for contradiction, that there exists \(n_{0}\) such that for all \(n>n_{0}\), \[\label{e6} (\mathbf{x}_{\mathrm{min}})^{p}<\frac{1}{n}\Big(1-\frac{p}{(p-1)s\log n}\Big).\tag{7}\] Set \(\lambda_{n}^{(p)}:=\lambda^{(p)}(Q,\mathcal{P}_{n})\). By Lemma 7, we can select sufficiently large \(n>n_{0}\) such that \[\frac{\lambda_{n-1}^{(p)}(n-1)^{s/p}}{(n-1)_{s}}-\frac{\lambda_{n}^{(p)}n^{s/p}}{(n)_{s}}<\frac{1}{n\log n}\cdot\frac{\lambda_{n}^{(p)}n^{s/p}}{(n)_{s}},\] and so \[\label{e7} \frac{\lambda_{n-1}^{(p)}}{\lambda_{n}^{(p)}}<\frac{n^{s/p-1}(n-s)}{(n-1)^{s/p}}\Big(1+\frac{1}{n\log n}\Big).\tag{8}\] Let \(k\in V(H_{n})\) be a vertex with \(x_{k}=\mathbf{x}_{\mathrm{min}}\). Then, by (4 ) and (8 ), we obtain \[\frac{1-sx_{k}^{p}}{(1-x_{k}^{p})^{s/p}}\leq \frac{\lambda_{n-1}^{(p)}}{\lambda_{n}^{(p)}} \leq \frac{n^{s/p-1}(n-s)}{(n-1)^{s/p}}\Big(1+\frac{1}{n\log n}\Big).\] Applying Fact 4 and (7 ), we derive \[\frac{1-\frac{s}{n}(1-\frac{p}{(p-1)s\log n})}{\Big({1-\frac{1}{n}(1-\frac{p}{(p-1)s\log n})}\Big)^{s/p}} \leq \frac{1-sx_{k}^{p}}{(1-x_{k}^{p})^{s/p}}\leq\frac{n^{s/p-1}(n-s)}{(n-1)^{s/p}}\Big(1+\frac{1}{n\log n}\Big),\] and hence \[\frac{\Big(n-s+\frac{p}{(p-1)\log n}\Big)n^{s/p-1}}{\Big(n-1+\frac{p}{(p-1)s\log n}\Big)^{s/p}} \leq\frac{n^{s/p-1}(n-s)}{(n-1)^{s/p}}\Big(1+\frac{1}{n\log n}\Big).\] This can be simplified to \[\label{subs} 1+\frac{p}{(p-1)(n-s)\log n}\leq\Big(1+\frac{p}{(p-1)s(n-1)\log n}\Big)^{s/p}\Big(1+\frac{1}{n\log n}\Big).\tag{9}\] For sufficiently large \(n\), we have \[\begin{align} \Big(1+\frac{p}{(p-1)s(n-1)\log n}\Big)^{s/p} &=1+\frac{1}{(p-1)(n-1)\log n}+O\Big(\frac{1}{(n\log n)^{2}}\Big)\\ &\leq1+\frac{1}{(p-1)(n-1)\log n}+\frac{1}{(p-1)(n-1)(n-2)\log n}\\ &=1+\frac{1}{(p-1)(n-2)\log n}. \end{align}\] Substituting this bound into (9 ), we obtain \[1+\frac{p}{(p-1)(n-s)\log n}\leq \Big(1+\frac{1}{(p-1)(n-2)\log n}\Big)\Big(1+\frac{1}{n\log n}\Big).\] By some cancellations and rearranging, we get \[\frac{p}{n-s}\leq \frac{1}{n-2}+\frac{p-1}{n}+\frac{1}{n(n-2)\log n}.\] Noting that \(\frac{p}{n-s}\geq\frac{p}{n-2}\), we have \[2(p-1)\leq\frac{1}{\log n},\] which leads to a contradiction for sufficiently large \(n\). ◻
Lemma 9. Let \(p>1\), and let \(Q\) be an \(s\)-vertex \(r\)-graph and \(H\) be an \(n\)-vertex \(r\)-graph with \(\lambda^{(p)}(Q,H)=\lambda\) and minimum \(Q\)-degree \(\delta\). Let \(\mathbf{x}\) be a principal \(Q\)-eigenvector corresponding to \(\lambda\). Then \[\Big(\frac{\lambda (\mathbf{x}_{\mathrm{min}})^{p-1}}{(s-1)!}\Big)^{p} \leq \frac{s!\binom{n}{s-1}\delta^{p-1}}{n^{s-1}}-(s!\tbinom{n}{s-1}\delta^{p-1}-\delta^{p})(\mathbf{x}_{\mathrm{min}})^{p(s-1)}.\]
Proof. Set \(V:=V(H)\), and let \(k\in V\) be a vertex achieving the minimum \(Q\)-degree \(\delta\). Considering the eigenequation for \(\lambda^{(p)}(Q,H)\) at vertex \(k\): \[\lambda (\mathbf{x}_{\mathrm{min}})^{p-1}\leq \lambda x^{p-1}_{k}=(s-1)!\sum_{I\in E_{Q,H}(k)}\mathcal{N}(Q,H[I])x_{I\backslash\{k\}}.\] By Hölder’s inequality, we have \[\label{e1} \bigg(\frac{\lambda (\mathbf{x}_{\mathrm{min}})^{p-1}}{(s-1)!}\bigg)^{p}\leq \delta^{p-1}\sum_{I\in E_{Q,H}(k)}\mathcal{N}(Q,H[I])(x_{I\backslash\{k\}})^{p}.\tag{10}\] Define \(T_{1}=\{I_{1}\in \binom{V}{s-1}:I_{1}\cup \{k\} \in E_{Q,H}(k)\}\) and \(T_{2}=\{I_{2}\in \binom{V}{s-1}:I_{2}\cup \{k\} \notin E_{Q,H}(k)\}\). Then \[\begin{align} \label{e2} \begin{aligned} \sum_{I\in E_{Q,H}(k)}\mathcal{N}(Q,H[I])(x_{I\backslash\{k\}})^{p} =&\sum_{I\in \binom{V}{s-1}}s!x_{I}^{p} -\sum_{I_{1}\in T_{1}}(s!-\mathcal{N}(Q,H[I_{1}\cup \{k\}]))x_{I_{1}}^{p}-\sum_{I_{2}\in T_{2}}s!x_{I_{2}}^{p}\\ \leq& \sum_{I\in \binom{V}{s-1}}s!x_{I}^{p} -\sum_{I_{1}\in T_{1}}(s!-\mathcal{N}(Q,H[I_{1}\cup \{k\}]))(\mathbf{x}_{\mathrm{min}})^{p(s-1)}\\ &-\sum_{I_{2}\in T_{2}}s!(\mathbf{x}_{\mathrm{min}})^{p(s-1)}\\ =&\sum_{I\in \binom{V}{s-1}}s!x_{I}^{p}-\big(s!\tbinom{n}{s-1}-\delta\big)(\mathbf{x}_{\mathrm{min}})^{p(s-1)}. \end{aligned} \end{align}\tag{11}\] By Maclaurin’s inequality, we have \[\label{e3} \sum_{I\in \binom{V}{s-1}}x_{I}^{p}\leq \binom{n}{s-1}\Bigg(n^{-1}\sum_{i\in V}x_{i}^{p}\Bigg)^{s-1}= \frac{\binom{n}{s-1}}{n^{s-1}}.\tag{12}\] The result is obtained by combining the inequalities (10 ), (11 ), and (12 ). ◻
Now we present the proof of Theorem 6.
Proof of Theorem 6. Observe that if \(\lambda^{(p)}(Q,\mathcal{P})=0\), then it follows from inequality (6 ) that \(\pi(Q,\mathcal{P})=0\).
Next, assume \(\lambda^{(p)}(Q,\mathcal{P})>0\). Suppose that \(H_{n}\in \mathcal{P}_{n}\) is an \(r\)-graph with \(\lambda^{(p)}(Q,H_{n})=\lambda^{(p)}(Q,\mathcal{P}_{n})=:\lambda\) and minimum \(Q\)-degree \(\delta\). Let \(\mathbf{x}=(x_1,\ldots,x_n)\) be a principal \(Q\)-eigenvector corresponding to \(\lambda^{(p)}(Q,H_{n})\). By Lemma 8, there exists an increasing infinite sequence \(\{n_{i}\}_{i=1}^{\infty}\) of positive integers such that for each \(n\in \{n_1,n_2,\ldots \}\), \[(\mathbf{x}_{\mathrm{min}})^{p}\geq\frac{1}{n}\Big(1-\frac{p}{(p-1)s\log n}\Big).\]
From Theorem 5 and Lemma 9, we derive \[\begin{align} \begin{aligned} \big(1-o(1)\big)\Big(\frac{\lambda^{(p)}(Q,\mathcal{P})}{(s-1)!}\Big)^{p}\cdot \frac{((n)_{s})^{p}}{n^{s+p-1}} &\leq\Big(\frac{\lambda (\mathbf{x}_{\mathrm{min}})^{p-1}}{(s-1)!}\Big)^{p}\\ &\leq \frac{s!\binom{n}{s-1}\delta^{p-1}}{n^{s-1}}-(s!\tbinom{n}{s-1}\delta^{p-1}-\delta^{p})(\mathbf{x}_{\mathrm{min}})^{p(s-1)}\\ &\leq \frac{s!\binom{n}{s-1}\delta^{p-1}}{n^{s-1}}-\frac{s!\binom{n}{s-1}\delta^{p-1}}{n^{s-1}}\big(1-o(1)\big)+\frac{\delta^{p}}{n^{s-1}}\\ &\leq o(n^{(s-1)(p-1)})+\frac{\delta^{p}}{n^{s-1}}. \end{aligned} \end{align}\]
Since \(\delta\leq \frac{s\mathcal{N}(Q,H_{n})}{n}\leq \frac{sex(Q,\mathcal{P}_{n})}{n}\), it follows that \[(1-o(1))(\lambda^{(p)}(Q,\mathcal{P}))^{p} \leq o(1)+\Big(\frac{ex(Q,\mathcal{P}_{n})}{\binom{n}{s}}\Big)^{p},\] where the term \(o(1)\) tends to \(0\) as \(n\to \infty\). Consequently, \[(\lambda^{(p)}(Q,\mathcal{P}))^{p}=\lim_{i \to \infty}(1-o(1))(\lambda^{(p)}(Q,\mathcal{P}))^{p}\leq \lim_{i \to \infty}o(1)+\Big(\frac{ex(Q,\mathcal{P}_{n_{i}})}{\binom{n_{i}}{s}}\Big)^{p}=(\pi(Q,\mathcal{P}))^{p}.\] Combining this inequality with (6 ) completes the proof of Theorem 6. ◻
Remark 10. The assumption that the hereditary property \(\mathcal{P}\) of \(r\)-graphs is closed under disjoint union with an isolated vertex is only used to ensure the monotonicity of \(\lambda^{(1)}(Q,\mathcal{P}_n)\) in \(n\). For the case \(p>1\), the proof of Theorem 6 does not rely on this condition. Consequently, the theorem remains valid for any hereditary property \(\mathcal{P}\) of \(r\)-graphs.
The celebrated Erdős-Stone-Simonovits theorem states that \[\mathrm{ex}(n,F)= \bigg(1-\frac{1}{\chi(F)-1}+o(1) \bigg)\frac{n^{2}}{2},\] where \(\chi(F)\) is the chromatic number of \(F\). In [10], Alon and Shikhelman extended the Erdős-Stone-Simonovits theorem to count copies of \(K_{s}\).
Lemma 11 ([10]). Let \(F\) be a graph with \(\chi(F)=k\). Then \[ex(K_{s},\overline{F}_{n})=\binom{k-1}{s}\bigg(\frac{n}{k-1}\bigg)^{s}+o(n^{s}).\]
Applying Theorem 6 and Lemma 11 yields the following result directly.
Corollary 12. Let \(p>1\) and \(F\) be a graph with \(\chi(F)=k\). Then \[\lambda^{(p)}(K_{s},\overline{F}_{n})=\frac{(k-1)_{s}}{(k-1)^{s}}n^{s-s/p}+o(n^{s-s/p}).\]
Remark 13. The case where \(p=s=2\) in Corollary 12 corresponds to the spectral Erdős-Stone-Simonovits theorem by Nikiforov [11]. Thus, Corollary 12 can be viewed as a generalization of adjacency spectral version of the Erdős-Stone-Simonovits theorem.
Gerbner and Palmer [12] gave a further extension to count arbitrary graphs \(H\) using the regularity lemma.
Lemma 14 ([12]). Let \(H\) be an \(s\)-vertex graph, and let \(F\) be a graph with \(\chi(F)=k\). Then \[ex(H,\overline{F}_{n})=ex(H, (\overline{K_{k}})_{n})+o(n^{s}).\]
Below, we present the spectral version of Lemma 14.
Corollary 15. Let \(p>1\), and let \(H\) be an \(s\)-vertex graph and \(F\) be a graph with \(\chi(F)=k\). Then \[\lambda^{(p)}(H,\overline{F}_{n})=\lambda^{(p)}(H, (\overline{ K_{k}})_{n})+o(n^{s-s/p}).\]
Proof. By Theorem 6, we have \[\frac{\lambda^{(p)}(H,\overline{F}_{n})}{n^{s-s/p}}=\big(1+o(1)\big)\frac{ex(H,\overline{F}_{n})}{\binom{n}{s}},\] and thus, \[\lambda^{(p)}(H,\overline{F}_{n})=s!ex(H,\overline{F}_{n})n^{-s/p}+o(n^{s-s/p}).\] Similarly, \[\lambda^{(p)}(H,(\overline{K_{k}})_{n})=s!ex(H,(\overline{K_{k}})_{n})n^{-s/p}+o(n^{s-s/p}).\] By Lemma 14, we obtain \[\begin{align} \begin{aligned} \lambda^{(p)}(H,\overline{F}_{n})-\lambda^{(p)}(H,(\overline{K_{k}})_{n})&= s!(ex(H,\overline{F}_{n})-ex(H,(\overline{K_{k}})_{n}))n^{-s/p}+o(n^{s-s/p})\\ &=o(n^{s-s/p}), \end{aligned} \end{align}\] which completes the proof. ◻
Consider an \(r\)-graph \(H\) on \(n\) vertices and a sequence of positive integers \(k_{1},\ldots,k_{n}\). The blow-up of \(H\) with respect to \(k_{1},\ldots,k_{n}\), denoted by \(H(k_{1},\ldots,k_{n})\), is the \(r\)-graph obtained by replacing each vertex \(i\in V(H)\) with a vertex class \(V_{i}\) (also called a block) of size \(k_{i}\), and if \(\{i_{1},\ldots,i_{r}\}\in E(H)\), then \(\{i_{1,j_{1}},\ldots,i_{r,j_{r}}\} \in E(H(k_{1},\ldots,k_{n}))\) for every \(i_{1,j_{1}}\in V_{i_{1}},\ldots,i_{r,j_{r}}\in V_{i_{r}}\). A property \(\mathcal{M}\) of \(r\)-graphs is multiplicative if \(H\in \mathcal{M}\) implies that every blow-up \(H(k_{1},\ldots,k_{n})\) is also in \(\mathcal{M}\) (i.e., \(\mathcal{M}\) is closed under the blow-up operation).
For an \(s\)-vertex \(r\)-graph \(Q\) and a hereditary property \(\mathcal{P}\) of \(r\)-graphs, we say that \(\mathcal{P}\) is Q-flat if \(\lambda^{(1)}(Q,\mathcal{P})=\pi(Q,\mathcal{P})\). We establish the following sufficient condition for \(Q\)-flat properties.
Lemma 16. Let \(Q\) be an \(r\)-graph and \(\mathcal{P}\) be a hereditary and multiplicative property of \(r\)-graphs. Then \(\mathcal{P}\) is \(Q\)-flat, and hence \(\lambda^{(p)}(Q,\mathcal{P})=\pi(Q,\mathcal{P})\) for every \(p\geq 1\).
Proof. By Theorem 6, we have \(\lambda^{(p)}(Q,\mathcal{P})=\pi(Q,\mathcal{P})\) for all \(p>1\). Inequality (6 ) gives \(\lambda^{(1)}(Q,\mathcal{P})\geq\pi(Q,\mathcal{P})\). It remains to show the reverse inequality \(\lambda^{(1)}(Q,\mathcal{P})\leq\pi(Q,\mathcal{P})\).
Consider \(H\in \mathcal{P}_{n}\) with \(\lambda^{(1)}(Q,H)=\lambda^{(1)}(Q,\mathcal{P}_{n})\), and let \(\mathbf{x}=(x_1,\ldots,x_n)\) be a principal \(Q\)-eigenvector corresponding to \(\lambda^{(1)}(Q,H)\) (with \(\|\mathbf{x}\|_{1}=1\)). We claim that \[\label{1f} \lambda^{(1)}(Q,H)=P_{Q,H}(\mathbf{x})\leq \pi(Q,\mathcal{P}).\tag{13}\] Since \(P_{Q,H}(\mathbf{x})\) is continuous in each variable, it suffices to prove inequality (13 ) for positive rational numbers \(x_{1},\ldots,x_{n}\). Thus, we can assume that \[x_{1}=k_{1}/k,\ldots,x_{n}=k_{n}/k,\] where \(k,k_{1},\ldots,k_{n}\) are positive integers and \(k=k_{1}+\cdots+k_{n}\). Consequently, inequality (13 ) is equivalent to \[\label{1f1f} \frac{P_{Q,H}((k_{1},\ldots,k_{n}))}{k^{s}}\leq \pi(Q,\mathcal{P}).\tag{14}\]
Let \(H(k_{1},\ldots,k_{n})\) be the blow-up of \(H\) with blocks \(V_{1},\ldots,V_{n}\), and let \(Q\) be an \(s\)-vertex \(r\)-graph. For any \(i_{1},\ldots,i_{s}\in V(H)\), the subgraph \(H(k_{1},\ldots,k_{n})[V_{i_{1}}\cup\ldots\cup V_{i_{s}}]\) contains at least \(k_{i_1}\times\cdots\times k_{i_s}\) copies of \(H[\{i_{1},\ldots,i_{s}\}]\). It follows that \[\mathcal{N}(Q,H(k_{1},\ldots,k_{n})[V_{i_{1}}\cup\ldots\cup V_{i_{s}}])\geq k_{i_{1}}\cdots k_{i_{s}}\mathcal{N}(Q,H[\{i_{1},\ldots,i_{s}\}]).\] This implies that \[\begin{align} \begin{aligned} P_{Q,H}((k_{1},\ldots,k_{n}))&=s!\sum_{\{i_{1},\ldots,i_{s}\}\in E(Q,H)}\mathcal{N}(Q,H[\{i_{1},\ldots,i_{s}\}])k_{i_{1}}\cdots k_{i_{s}}\\ &\leq s!\sum_{\{i_{1},\ldots,i_{s}\}\in E(Q,H)}\mathcal{N}(Q,H(k_{1},\ldots,k_{n})[V_{i_{1}}\cup\ldots\cup V_{i_{s}}])\\ &\leq P_{Q,H(k_{1},\ldots,k_{n})}((1,\ldots,1))=s!\mathcal{N}(Q,H(k_{1},\ldots,k_{n})). \end{aligned} \end{align}\] Since \(H(k_{1},\ldots,k_{n})\in\mathcal{P}\) (as \(\mathcal{P}\) is multiplicative) and \(\nu(H(k_{1},\ldots,k_{n}))=k\), we obtain \[\frac{s!\mathcal{N}(Q,H(k_{1},\ldots,k_{n}))}{k^{s}}\leq \frac{ex(Q,\mathcal{P}_{k})}{\binom{k}{s}} \leq \pi(Q,\mathcal{P})+o(1),\] where the term \(o(1)\) tends to \(0\) as \(k\to \infty\).
Similarly, for every positive integer \(t\), we have \[\begin{align} \begin{aligned} \frac{P_{Q,H}((k_{1},\ldots,k_{n}))}{k^{s}} &=\frac{P_{Q,H}((t k_{1},\ldots,t k_{n}))}{(t k)^{s}}\\ &\leq \frac{s!\mathcal{N}(Q,H(tk_{1},\ldots,tk_{n}))}{(t k)^{s}}\\ &\leq \pi(Q,\mathcal{P})+o(1). \end{aligned} \end{align}\] Taking \(t\to \infty\), we establish inequality (14 ), hence inequality (13 ) holds. Therefore, \[\lambda^{(1)}(Q,\mathcal{P})=\lim\limits_{n\to \infty}\lambda^{(1)}(Q,\mathcal{P}_{n})\leq\pi(Q,\mathcal{P}),\] completing the proof. ◻
Notably, the complete graph \(K_{l+1}\) is \(2\)-covering, which implies that \(\overline{K_{l+1}}\) is \(K_{2}\)-flat. A classical theorem of Turán [13] establishes that for any \(K_{l+1}\)-free graph \(G\) on \(n\) vertices, the number of edges satisfies \(e(G)\leq (1-\frac{1}{l})\frac{n^{2}}{2}\). Wilf [14] later provided a spectral extension of Turán’s theorem, demonstrating that if \(G\) is an \(n\)-vertex \(K_{l+1}\)-free graph, then its largest eigenvalue \(\lambda(G)\) satisfies \(\lambda(G)\leq(1-\frac{1}{l})n\). In 2002, Nikiforov [15] further extended this result by proving that for any \(K_{l+1}\)-free graph \(G\) with \(m\) edges, \(\lambda(G)\leq(1-\frac{1}{l})^{1/2}(2m)^{1/2}\). In the following, we generalize these bounds to families of \(r\)-graphs with \(Q\)-flat properties.
Theorem 17. If \(Q\) is an \(s\)-vertex \(r\)-graph and \(\mathcal{P}\) is a \(Q\)-flat property of \(r\)-graphs, then for any \(H\in \mathcal{P}_{n}\), \[\mathcal{N} (Q,H)\leq\pi(Q,\mathcal{P})n^{s}/s!,\] and for every \(p\geq1\), \[\lambda^{(p)}(Q,H)\leq\pi(Q,\mathcal{P})n^{s-s/p}.\]
Proof. Let \(\mathbf{x}=(x_1,\ldots,x_n)\) be a \(Q\)-principal eigenvector corresponding to \(\lambda^{(p)}(Q,H)\) with \(\|\mathbf{x}\|_{p}=1\). Then \[\frac{\lambda^{(p)}(Q,H)}{n^{s-s/p}}= \frac{P_{Q,H}(\mathbf{x})}{n^{s-s/p}} =P_{Q,H}((x_1/n^{1-1/p},\ldots,x_n/n^{1-1/p})).\] By Power-Mean inequality, for any \(p\geq1\), \[\frac{x_1+\cdots+x_n}{n^{1-1/p}}\leq(x_1^p+\cdots+x_n^p)^{1/p}=1.\] From Theorem 5, it follows that \[\frac{\lambda^{(p)}(Q,H)}{n^{s-s/p}}\leq \lambda^{(1)}(Q,H)\leq \lambda^{(1)}(Q,\mathcal{P}_{n})\leq\lambda^{(1)}(Q,\mathcal{P})=\pi(Q,\mathcal{P}).\] Since \(\lambda^{(p)}(Q,H)\geq s!\mathcal{N}(Q,H)/n^{s/p}\), we conclude \[\mathcal{N}(Q,H)\leq\pi(Q,\mathcal{P})n^{s}/s!,\] completing the proof. ◻
Lemma 18. If \(Q\) is an \(s\)-vertex \(r\)-graph and \(\mathcal{P}\) is a \(Q\)-flat property of \(r\)-graphs, then for any \(p\geq1\) and \(H\in \mathcal{P}\), \[\lambda^{(p)}(Q,H)\leq\pi(Q,\mathcal{P})^{1/p}(s!\mathcal{N}(Q,H))^{1-1/p}.\]
Proof. Lemma 2 implies that \[\big(\lambda^{(p)}(Q,H)\big)^{p}\leq\lambda^{(1)}(Q,H)(s!\mathcal{N}(Q,H))^{p-1}.\] Moreover, since \(\mathcal{P}\) is \(Q\)-flat, Theorem 5 yields \[\lambda^{(1)}(Q,H)\leq\lambda^{(1)}(Q,\mathcal{P})=\pi(Q,\mathcal{P}).\] Combining these inequalities, we obtain \[\lambda^{(p)}(Q,H)\leq\pi(Q,\mathcal{P})^{1/p}(s!\mathcal{N}(Q,H))^{1-1/p},\] completing the proof. ◻
In this section, we study spectral generalized Turán problems for a family of \(\mathcal{F}\)-free \(r\)-graphs. We assume that all members of \(\mathcal{F}\) contain no isolated vertices.
The following spectral stability theorem indicates that if the maximum \((p,Q)\)-spectral radius over all \(\mathcal{F}\)-free \(r\)-graphs satisfies a specific growth condition, then the extremal hypergraphs must have a large minimum \(Q\)-degree.
Theorem 19. Let \(p>1\), \(s\geq r\geq2\), and \(0<\varepsilon<1\). Let \(Q\) be an \(s\)-vertex \(r\)-graph, and let \(\mathcal{F}\) be a family of \(r\)-graphs with \(\pi(Q,\overline{\mathcal{F}})>0\). Let \(\mathcal{G}_n\) be the collection of all \(n\)-vertex \(\mathcal{F}\)-free \(r\)-graphs with minimum \(Q\)-degree at least \((1-\varepsilon)\pi(Q,\overline{\mathcal{F}})\binom{n}{s-1}\) and define \(\lambda^{(p)}(Q,\mathcal{G}_{n})=\max\{\lambda^{(p)}(Q,G): G\in \mathcal{G}_{n}\}\). Suppose that there exists a sufficiently large \(n_{0}\in \mathbb{N}\) such that for every \(n\geq n_{0}\), we have \[\label{d12d} \lambda^{(p)}(Q,\overline{\mathcal{F}}_{n}) \ge \lambda^{(p)}(Q,\overline{\mathcal{F}}_{n-1}) + \pi(Q,\overline{\mathcal{F}})(s-s/p) (1-\sigma)n^{s-s/p-1},\qquad{(1)}\] where \(\sigma=\varepsilon \pi(Q,\overline{\mathcal{F}})/(5s!(s-1)).\) Then for any \(\mathcal{F}\)-free \(r\)-graph \(H\) on \(n\geq n_{0}\) vertices, we have \[\lambda^{(p)}(Q,H)\leq \lambda^{(p)}(Q,\mathcal{G}_{n}).\] In addition, if the equality holds, then \(H\in\mathcal{G}_{n}\).
We require the following Lemma for the proof of Theorem 19.
Lemma 20. Let \(p>1\), \(s\geq r\geq2\), and \(0<\varepsilon<1\). Let \(Q\) be an \(s\)-vertex \(r\)-graph, and let \(\mathcal{P}\) be a hereditary property of \(r\)-graphs with \(\pi(Q,\mathcal{P})>0\). Let \(H_{n}\in \mathcal{P}_{n}\) satisfy \(\lambda^{(p)}(Q,H_{n})=\lambda^{(p)}(Q,\mathcal{P}_{n})\). Suppose \(0\leq\varepsilon'<\varepsilon \pi(Q,\mathcal{P})/(s!(s-1))\), and let \(\mathbf{x}\) be a principal \(Q\)-eigenvector corresponding to \(\lambda^{(p)}(Q,H_{n})\). If \(n\) is sufficiently large and \((\mathbf{x}_{\mathrm{min}})^{p}\geq\frac{1-\varepsilon'}{n},\) then \[\delta_{Q}(H_{n})\geq(1-\varepsilon)\pi(Q,\mathcal{P})\binom{n}{s-1}.\]
Proof. Set \(\delta:=\delta_{Q}(H_{n})\) and \(\lambda:=\lambda^{(p)}(Q,H_{n})\). Suppose for contradiction that \(\delta<(1-\varepsilon)\pi(Q,\mathcal{P})\binom{n}{s-1}\). By Theorem 5 and Lemma 9, we obtain \[\begin{align} \begin{aligned} (1-\varepsilon')^{p-1}\Big(\frac{\lambda^{(p)}(Q,\mathcal{P})}{(s-1)!}\Big)^{p}\cdot \frac{((n)_{s})^{p}}{n^{s+p-1}} & \leq\Big(\frac{\lambda (\mathbf{x}_{\mathrm{min}})^{p-1}}{(s-1)!}\Big)^{p}\\ &\leq \frac{s!\binom{n}{s-1}\delta^{p-1}}{n^{s-1}}-(s!\tbinom{n}{s-1}\delta^{p-1}-\delta^{p})(\mathbf{x}_{\mathrm{min}})^{p(s-1)}\\ &\leq \frac{s!\binom{n}{s-1}\delta^{p-1}}{n^{s-1}}-\frac{1}{n^{s-1}}(s!\tbinom{n}{s-1}\delta^{p-1}-\delta^{p})(1-(s-1)\varepsilon')\\ &\leq \frac{s!(s-1)\varepsilon'\binom{n}{s-1}\delta^{p-1}}{n^{s-1}}+\frac{\delta^{p}}{n^{s-1}}\\ &\leq \frac{\binom{n}{s-1}\delta^{p-1}}{n^{s-1}}(s!(s-1)\varepsilon'+ (1-\varepsilon)\pi(Q,\mathcal{P}))\\ &\leq \frac{\pi(Q,\mathcal{P})\binom{n}{s-1}\delta^{p-1}}{n^{s-1}}, \end{aligned} \end{align}\] where the last inequality follows from \(\varepsilon'<\varepsilon \pi(Q,\mathcal{P})/(s!(s-1))\). By Theorem 6 and the assumption \(\delta<(1-\varepsilon)\pi(Q,\mathcal{P})\binom{n}{s-1}\), we further get \[(1-\varepsilon')^{p-1}\big(\pi(Q,\mathcal{P})\tbinom{n-1}{s-1}\big)^{p}\leq(1-\varepsilon)^{p-1}\big(\pi(Q,\mathcal{P})\tbinom{n}{s-1}\big)^{p},\] which implies that \[1<\Big(\frac{1-\varepsilon'}{1-\varepsilon}\Big)^{\frac{p-1}{p}}\leq \frac{n}{n-s+1}.\] This is a contradiction for sufficiently large \(n\), completing the proof. ◻
Proof of Theorem 19. Let \(H_{n}\) be an \(\mathcal{F}\)-free \(r\)-graph on \(n\) vertices that satisfies \(\lambda^{(p)}(Q,H_{n})=\lambda^{(p)}(Q,\overline{\mathcal{F}}_{n})\) , and let \(\mathbf{x}=(x_{1},\ldots,x_{n})\) be a principal \(Q\)-eigenvector corresponding to \(\lambda^{(p)}(Q,H_{n})\). In view of Lemma 20, it suffices to show that for \(n\geq n_{0}\), \[(\mathbf{x}_{\mathrm{min}})^{p}\geq\frac{1-\varepsilon'}{n},\] where \(\varepsilon'=\varepsilon \pi(Q,\overline{\mathcal{F}})/(2s!(s-1))\). Suppose for contradiction that for some \(n\), \[(\mathbf{x}_{\mathrm{min}})^{p}<\frac{1-\varepsilon'}{n}.\] Applying (4 ), Fact 4, and Bernoulli’s inequality, we obtain \[\begin{align} \begin{aligned} \frac{\lambda^{(p)}(Q,H_{n-1})}{\lambda^{(p)}(Q,H_{n})} &\geq\frac{1-s(\mathbf{x}_{\mathrm{min}})^{p}}{(1-(\mathbf{x}_{\mathrm{min}})^{p})^{s/p}}\\ &\geq\Big(1-\frac{s(1-\varepsilon')}{n}\Big)\Big(1-\frac{1-\varepsilon'}{n}\Big)^{-s/p}\\ &\geq\Big(1-\frac{s(1-\varepsilon')}{n}\Big)\Big(1+\frac{s(1-\varepsilon')}{pn}\Big)\\ &= 1-\frac{(s-s/p)(1-\varepsilon')}{n}-\frac{s^{2}(1-\varepsilon')^{2}}{pn^{2}}. \end{aligned} \end{align}\] From Theorem 6, it follows that \(\lambda^{(p)}(Q,H_{n})= (\pi(Q,\overline{\mathcal{F}})+o(1))n^{s-s/p}\), and hence \[\label{k1} \lambda^{(p)}(Q,H_{n-1})\geq \lambda^{(p)}(Q,H_{n})-\pi(Q,\overline{\mathcal{F}})(s-s/p)(1-\varepsilon'/2)n^{s-s/p-1}.\tag{15}\]
On the other hand, by (?? ), we have \[\label{k2} \lambda^{(p)}(Q,H_{n})\geq \lambda^{(p)}(Q,H_{n-1})+\pi(Q,\overline{\mathcal{F}})(s-s/p)(1-\sigma)n^{s-s/p-1}.\tag{16}\] Combining (15 ) and (16 ) yields \[\pi(Q,\overline{\mathcal{F}})(s-s/p)(1-\sigma)n^{s-s/p-1}\leq \pi(Q,\overline{\mathcal{F}})(s-s/p)(1-\varepsilon'/2)n^{s-s/p-1},\] which contradicts \(\sigma=\varepsilon \pi(Q,\overline{\mathcal{F}})/(5s!(s-1))\). This completes the proof of Theorem 19. ◻
In \(1984\), Erdős conjectured that for every \(n\geq5\), the balanced blow-up of \(C_{5}\) contains the maximum number of copies of \(C_{5}\) among all \(n\)-vertex triangle-free graphs. This conjecture was first resolved independently by Grzesik [16] and Hatami et al. [17] for sufficiently large \(n\). Later, Lidický and Pfender [18] completed the proof by extending the result to all \(n\).
Lemma 21 ([18]). For all \(n\), the maximum number of copies of \(C_{5}\) in \(K_{3}\)-free graphs on \(n\) vertices is \[\prod^{4}_{i=0}\Big\lfloor\frac{n+i}{5}\Big\rfloor.\] Moreover, for \(n\geq9\), the only \(K_{3}\)-free graph on \(n\) vertices maximizing the number of copies of \(C_{5}\) is the balanced blow-up of \(C_{5}\).
Lemma 22 ([19]). There exist \(\varepsilon>0\) and \(N_{0}\) such that the following holds for all \(n\geq N_{0}\). If \(G\) is an \(n\)-vertex \(K_{3}\)-free graph with \(\delta_{C_{5}}(G)\geq (1/5^{4}-\varepsilon)n^{4}\), then \(G\) is \(C_{5}\)-colorable.
Remark 23. We remark that the minimum \(Q\)-degree in [19] differs from our by a constant factor of \(|Aut(Q)|\). Additionally, observe that \(|Aut(C_{5})|=10\).
Lemma 24 ([20]). Let \(l\geq r\geq2\). Then \(e(T^{r}_{l}(n))=\frac{(l)_{r}}{r!l^{r}}n^{r}+O(n^{r-2}).\)
Lemma 25 ([21]). Let \(l\geq r\geq2\), and let \(G\) be an \(l\)-partite \(r\)-graph of order \(n\). For every \(p>1\), \[\lambda^{(p)}(G)\leq \lambda^{(p)}(T^{r}_{l}(n)),\] with equality if and only if \(G=T^{r}_{l}(n)\).
For any \(r\)-graph \(Q\) on \(s\) vertices and any \(r\)-graph \(H\), we define \(D(Q,H)\) as the \(s\)-graph derived from \(H\) with vertex set \(V(D(Q,H))=V(H)\) and edge set \[E(D(Q,H))=\{\{v_{1},\ldots,v_{s}\}:H[v_{1},\ldots,v_{s}]\supseteq Q\}.\] Note that if \(\mathcal{N}(Q,H[v_{1},\ldots,v_{s}])=1\) for any \(\{v_{1},\ldots,v_{s}\}\in E(D(Q,H))\), then \[\label{t10} \mathcal{N}(Q,H)=e(D(Q,H))~~and~~\lambda^{(p)}(Q,H)=\lambda^{(p)}(D(Q,H)).\tag{17}\]
Recently, Liu [1] established a general theorem that extends the result of Keevash-Lenz-Mubayi and applied it to obtain a spectral Erdős pentagon theorem. We extend Liu’s result via a different approach. For any \(p\geq1\), an \(r\)-graph \(Q\) and a family \(\mathcal{G}_{n}\) of \(r\)-graphs on \(n\) vertices, let \(\lambda^{(p)}(Q,\mathcal{G}_{n})\) (resp. \(\lambda^{(p)}(\mathcal{G}_{n})\)) denote the maximum \((p,Q)\)-spectral radius (resp. \(p\)-spectral radius) among all \(r\)-graphs in \(\mathcal{G}_{n}\).
Theorem 26. Let \(p\geq 1\), and let \(\mathcal{L}_{n}\) be the balanced blow-up of \(C_{5}\) on \(n\) vertices. Then, for all sufficiently large \(n\) and any \(n\)-vertex \(K_{3}\)-free graph \(G\), we have \(\lambda^{(p)}(C_{5},G)\leq \lambda^{(p)}(C_{5},\mathcal{L}_{n})\). The equality holds if and only if \(G=\mathcal{L}_{n}\) for \(p>1\), and if \(G\supseteq C_{5}\) for \(p=1\).
Proof. For any \(C_{5}\)-colorable graph \(H\) with a homomorphism \(\phi\) from \(V(H)\) to \(V(C_{5})\), denote \(V(C_{5})\) as \(\{1,2,3,4,5\}\). For each \(i\in[5]\), define \(V_{i}=\{v\in V(H): \phi(v)=i\}\) (some \(V_{i}\) may be empty). This defines a natural partition of \(V(H)\).
We claim that for any five vertices \(v_{1},\ldots,v_{5}\) in \(V(H)\), if \(H[v_{1},\ldots,v_{5}]\) contains a copy of \(C_{5}\) (in fact, \(H[v_{1},\ldots,v_{5}]\cong C_{5}\)), then these five vertices must belong to five distinct parts in the partition.
Suppose \(v_{1}v_{2}v_{3}v_{4}v_{5}v_{1}\) forms a copy of \(C_{5}\) in \(H[v_{1},\ldots,v_{5}]\), with \(\{v_{i}v_{i+1}\}\in E(H)\) for \(i\in [5]\) (indices modulo \(5\)). Assume for contradiction that the claim fails. By symmetry, we may assume that \(v_{1}\) and \(v_{3}\) belong to the same part, i.e., \(\phi(v_{1})=\phi(v_{3})\). Since \(H\) is \(C_{5}\)-colorable and \(\{v_{5}v_{1}\} \in E(H)\), it follows that \[\{\{\phi(v_{5})\phi(v_{3})\},\{\phi(v_{3})\phi(v_{4})\}, \{\phi(v_{4})\phi(v_{5})\}\}\subseteq E(C_{5}).\] which contradicts the fact that \(C_{5}\) is \(K_{3}\)-free. Therefore, the claim holds.
The above claim implies that \(D(C_{5},H)\) is a \(5\)-partite \(5\)-graph. It follows that \(D(C_{5},\mathcal{L}_{n})\) is isomorphic to \(T_{5}^{5}(n)\), and hence \(\mathcal{N}(C_{5},\mathcal{L}_{n})=e(T_{5}^{5}(n))\). By Lemmas 21 and 24, we have \[\label{exe} ex(C_{5},(\overline{K_{3}})_{n})=\mathcal{N}(C_{5},\mathcal{L}_{n})=e(T_{5}^{5}(n))=\frac{n^{5}}{5^{5}}+O(n^{3}),\tag{18}\] and hence \(\pi(C_{5},\overline{K_{3}})= 5!/5^{5}\).
Let \(\mathrm{Col}(C_{5})_{n}\) be the set of all \(C_{5}\)-colorable graphs on \(n\) vertices, and let \[\mathcal{R}_{n}:=\{D(C_{5},H): H\in \mathrm{Col}(C_{5})_{n}\}.\] Then, by (17 ), \[\label{t11} \lambda^{(p)}(C_{5},\mathrm{Col}(C_{5})_{n})=\lambda^{(p)}(\mathcal{R}_{n}).\tag{19}\] Lemma 22 shows that there exist \(\varepsilon>0\) and \(N_{0}\) such that for every \(n\)-vertex \(K_{3}\)-free graph \(G\) with \(\delta_{C_{5}}(G)\geq(1/5^{4}-\varepsilon)n^{4}\) is contained in \(\mathrm{Col}(C_{5})_{n}\).
By (5 ) and (18 ), we have \[\label{1k3} \lambda^{(p)}(C_{5},(\overline{K_{3}})_{n})\geq 5!ex(C_{5},(\overline{K_{3}})_{n})/n^{5/p}\geq \pi(C_{5},\overline{K_{3}})n^{5-5/p}+O(n^{3-5/p}).\tag{20}\] Note that \(K_{3}\) is a \(2\)-covering graph. Lemma 16 and Theorem 17 imply that \[\begin{align} \label{2k3} \begin{aligned} \lambda^{(p)}(C_{5},(\overline{K_{3}})_{n-1}) &\leq \pi(C_{5},\overline{K_{3}})(n-1)^{5-5/p}\\ &=\pi(C_{5},\overline{K_{3}})n^{5-5/p}- \pi(C_{5},\overline{K_{3}})(5-5/p)n^{4-5/p}+O(n^{3-5/p}). \end{aligned} \end{align}\tag{21}\] Combining (20 ) and (21 ) yields \[\lambda^{(p)}(C_{5},(\overline{K_{3}})_{n})-\lambda^{(p)}(C_{5},(\overline{K_{3}})_{n-1}) \geq \pi(C_{5},\overline{K_{3}})(5-5/p)n^{4-5/p}+o(n^{4-5/p}).\] Thus, by Theorem 19 and equality (19 ), for \(p>1\) and enough large \(n\), we have \[\lambda^{(p)}(C_{5},(\overline{K_{3}})_{n})\leq \lambda^{(p)}(C_{5},\mathrm{Col}(C_{5})_{n})=\lambda^{(p)}(\mathcal{R}_{n})\leq\lambda^{(p)}(T_{5}^{5}(n))= \lambda^{(p)}(C_{5},\mathcal{L}_{n}),\] where the third inequality follows from Lemma 25.
For \(p=1\), by Theorem 17, we have \(\lambda^{(1)}(C_{5},(\overline{K_{3}})_{n})\leq\pi(C_{5},\overline{K_{3}})= 5!/5^{5}\). Moreover, observe that \[\lambda^{(1)}(C_{5},G)\geq\lambda^{(1)}(C_{5},C_{5})= \lambda^{(1)}(K^{5}_{5})=5!/5^{5},\] which implies \(\lambda^{(1)}(C_{5},G)=\lambda^{(1)}(C_{5},(\overline{K_{3}})_{n})\), completing the proof. ◻
For a graph \(H\) and \(e\in E(H)\), let \(H-e\) denote the graph with vertex set \(V(H)\) and edge set \(E(H)\backslash\{e\}\). A graph \(H\) is called edge-critical if there exists an edge \(e\) of \(H\) such that \(\chi(H-e)=\chi(H)-1\). The \(s\)-expansion \(H^{(s)}\) of \(H\) is the \(s\)-graph obtained from \(H\) by enlarging each edge of \(H\) with \(s-2\) new vertices disjoint from \(V(H)\) such that distinct edges of \(H\) are enlarged by distinct vertices.
Simonovits [22] extended Turán’s theorem to any edge-critical graph \(F\) and established the critical edge theorem. Later, Ma and Qiu [2] generalized Simonovit’s result as follows:
Theorem 27 ([2]). Let \(l\geq s\geq 2\) , and let \(F\) be an edge-critical graph with \(\chi(F)=l+1\). Then for sufficiently large \(n\), the unique \(n\)-vertex \(F\)-free graph with the maximum number of copies of \(K_{s}\) is the Turán graph \(T_{l}(n)\).
Very recently, Zheng, Li and Su [9] determined the maximum \(p\)-spectral radius among all \(n\)-vertex \(F^{(r)}\)-free \(s\)-graphs, where \(F\) is an edge-critical graph.
Lemma 28 ([9]). Let \(p\geq1\), \(l\geq s\geq 2\) , and let \(F\) be an edge-critical graph with \(\chi(F)=l+1\). Then there exists \(n_{0}\), such that for any \(F^{(s)}\)-free \(s\)-graph \(G\) on \(n>n_{0}\) vertices, \(\lambda^{(p)}(G)\leq\lambda^{(p)}(T^{s}_{l}(n))\). The equality holds if and only if \(G=T^{s}_{l}(n)\) for \(p>1\), and if \(G\supseteq K^{s}_{l}\) for \(p=1\).
Let \(H\) be an \(r\)-graph. The \(2\)-shadow of \(H\), denoted by \(\partial_{2}H\), is the graph with vertex set \(V(\partial_{2}H)=V(H)\) and edge set \(E(\partial_{2}H)=\{\{v_{1},v_{2}\}: \{v_{1},v_{2}\}\subseteq e\in E(H)\}\).
We present a spectral analogue of Theorem 27.
Theorem 29. Let \(p\geq1\), \(l\geq s\geq 2\) , and let \(F\) be an edge-critical graph with \(\chi(F)=l+1\). Then there exists \(n_{0}\), such that for any \(F\)-free graph \(G\) on \(n>n_{0}\) vertices, \(\lambda^{(p)}_{s}(G)\leq\lambda^{(p)}_{s}(T_{l}(n))\). The equality holds if and only if \(G=T_{l}(n)\) for \(p>1\), and if \(G\supseteq K_{l}\) for \(p=1\).
Proof. We define the following sets for a given integer \(n\): \[\mathcal{A}_{n}:=\{D(K_{s},G):G ~~is~~F-free~~ graph~~on~~n~~vertices\},\] \[\mathcal{B}_{n}:=\{H:H~~is~~an~~s-graph~~on~~n~~vertices~~and~~\partial_{2}H~~is~~F-free\},\] \[\mathcal{C}_{n}:=\{H:H~~is~~an~~s-graph~~on~~n~~vertices~~and~~H~~is~~F^{(s)}-free\}.\] Observe that for any \(F\)-free graph \(G\), we have \(\partial_{2}D(K_{s},G)\subseteq G\), which implies \(\mathcal{A}_{n}\subseteq \mathcal{B}_{n}\subseteq \mathcal{C}_{n}\). By Lemma 28, for \(p\geq1\), it follows that \[\lambda^{(p)}(\mathcal{A}_{n})\leq\lambda^{(p)}(\mathcal{C}_{n})=\lambda^{(p)}(T^{s}_{l}(n)).\] Note that \(D(K_{s},T_{l}(n))=T^{s}_{l}(n)\). From equality (17 ), we obtain \[\lambda^{(p)}(K_{s},\overline{F}_{n})=\lambda^{(p)}(\mathcal{A}_{n})=\lambda^{(p)}_{s}(T_{l}(n)).\] The result follows from Lemma 28. ◻
Remark 30. From Theorem 29, letting \(p\to \infty\) and applying Proposition 1 directly yields Theorem 27, but does not yield the uniqueness of the extremal graph. Moreover, Theorem 29 can be viewed as a generalization of the result of Yu and Peng [8].
In this paper, we systematically investigate the \((p,Q)\)-spectral radius of hypergraphs and derive several results concerning spectral generalized Turán problems. Specifically, Theorem 19 establishes a spectral stability result. We conjecture that the conclusion of Theorem 19 holds even without condition (?? ), leading to the following:
Conjecture 31. Let \(p>1\), \(s\geq r\geq2\), \(Q\) be an \(s\)-vertex \(r\)-graph and \(\mathcal{F}\) be a family of \(r\)-graphs with \(\pi(Q,\overline{\mathcal{F}})>0\). Let \(\mathcal{G}_n\) be the collection of all \(n\)-vertex \(\mathcal{F}\)-free \(r\)-graphs with minimum \(Q\)-degree more than \((1-\varepsilon)\pi(Q,\overline{\mathcal{F}})\binom{n}{s-1}\) and \(\lambda^{(p)}(Q,\mathcal{G}_{n})=\max\{\lambda^{(p)}(Q,G): G\in \mathcal{G}_{n}\}\). Then for any \(\mathcal{F}\)-free graph \(H\) on \(n\geq n_{0}\) vertices, we have \[\lambda^{(p)}(Q,H)\leq \lambda^{(p)}(Q,\mathcal{G}_{n}).\] In addition, if the equality holds, then \(H\in\mathcal{G}_{n}\).
To address this conjecture, we propose two potential approaches.
Problem 32. Let \(Q\) be an \(r\)-graph on \(s\) vertices, and \(\mathcal{P}\) be a hereditary property of \(r\)-graphs with \(\pi(Q,\mathcal{P})>0\). Suppose that \(H_{n}\in \mathcal{P}_{n}\) is an \(r\)-graph satisfying \(\lambda^{(p)}(Q,H_{n})=\lambda^{(p)}(Q,\mathcal{P}_{n})\) for \(p>1\) and \(\mathbf{x}=(x_1,\ldots,x_n)\) is a principal \(Q\)-eigenvector for \(\lambda^{(p)}(Q,H_{n})\). Does there exist a constant \(n_{0}\) such that for all \(n\geq n_{0}\), \[(\mathbf{x}_{\mathrm{min}})^{p}\geq\frac{1}{n}\Big(1-\frac{p}{(p-1)s\log n}\Big)?\] An affirmative answer to Problem 32 would, via Lemma 20, imply conjecture 31.
Problem 33. Let \(Q\) be an \(s\)-vertex \(r\)-graph, and let \(\mathcal{F}\) be a family of \(r\)-graphs with \(\pi(Q,\overline{\mathcal{F}})>0\). For \(p>1\), does there exist a sequence \(\{a_{n}\}\) such that \[\lambda^{(p)}(Q,\overline{\mathcal{F}}_{n})=\pi(Q,\overline{\mathcal{F}})n^{s-s/p}+a_{n}n^{(s-1)(1-1/p)},\] and the limit \(\lim\limits_{n\to \infty} a_{n}\) exists? If answered affirmatively, then \[\lambda^{(p)}(Q,\overline{\mathcal{F}}_{n})=\lambda^{(p)}(Q,\overline{\mathcal{F}}_{n-1}) + \pi(Q,\overline{\mathcal{F}})(s-s/p) n^{s-s/p-1}+o(n^{s-s/p-1}),\] and conjecture 31 would follow from Theorem 19.
Moreover, Problem 33 is of independent interest. Generally speaking, to what structural parameters of the graph is \(\{a_{n}\}\) related and for which hereditary families does the limit of \(\{a_{n}\}\) exist?
For any \(r\)-graph \(Q\) on \(s\) vertices and any \(r\)-graph \(H\), recall the definition of \(s\)-graph \(D(Q,H)\): its edge set is defined as \[E(D(Q,H))=\{\{v_{1},\ldots,v_{s}\}:H[v_{1},\ldots,v_{s}]\supseteq Q\}.\] Assigning a weight \(\mathcal{N}(Q,H[v_{1},\ldots,v_{s}])\) to each edge \(\{v_{1},\ldots,v_{s}\}\), we define its \(p\)-spectral radius as: \[\lambda^{(p)}(D(Q,H))=\max_{\|\mathbf{x}\|_{p}=1}s!\sum_{\{i_{1},\ldots,i_{s}\}\in E(D(Q,H))}\mathcal{N}(Q,H[\{i_{1},\ldots,i_{s}\}])x_{i_{1}}\cdots x_{i_{s}}.\] Then, it follows that \[\lambda^{(p)}(Q,H)=\lambda^{(p)}(D(Q,H)).\] This establishes a connection between the \((p,Q)\)-spectral radius of the \(r\)-graph \(H\) and the \(p\)-spectral radius of the weighted \(s\)-graph \(D(Q,H)\). For relevant conclusions regarding the \(p\)-spectral radius of weighted hypergraphs, one may refer to the results of Nikiforov [23], such as the Perron-Frobenius theory for the weighted hypergraphs discussed in Section \(5\) of [23].
No data was used for the research described in the article.