arXiv subjects

Bo Ning

Publications and source records attributed to Bo Ning.

At least 37 records · Page 2Linked to original sources

arXiv2026

On spectral Turán theorems: confirming a conjecture of Guiduli and two problems of Nikiforov

Let $G$ be an $n$-vertex graph, and let $λ(G)$ and $λ_n(G)$ denote the largest and smallest eigenvalues of its adjacency matrix. Write $e(G)$ for the number of edges of $G$, $d(G)=2e(G)/n$ for its average degree, and $T_r(n)$ for the $r$-partite Turán graph on $n$ vertices. We prove four sharp results in spectral Turán theory. First, we confirm Guiduli's spectral dense-neighborhood conjecture (1996) in a stronger form: if $λ(G)\ge λ(T_r(n))$, then either $G\cong T_r(n)$, or there exists a vertex $v$ such that $λ(G[N(v)]) > λ(T_{r-1}(d(v)))$. Moreover, when $λ(G)>λ(T_r(n))$, every vertex attaining the maximum entry in any nonnegative Perron eigenvector of $G$ has this property. Second, we answer a problem of Nikiforov (2009) by showing that the exact Turán edge threshold is detected by the exact spectral threshold: for every $r\ge 2$ and every $n$, $λ(G)<λ(T_r(n))$, implying $e(G)<e(T_r(n)).$ Our proof also determines the equality cases. Third, we answer another question of Nikiforov (2009) by showing that his least-eigenvalue clique bound \[ ω(G)\ge 1+\frac{2e(G)}{(n-d(G))(d(G)-λ_n(G))} \] does imply the concise form of Turán's theorem. Finally, we discuss an open problem proposed by Ai et al. (2026) in \cite{ALNS26+}.

math.CO↗

arXiv2026

An exponentially small gap of the Perron vector on independent sets

A classical result of Cioabă states that if $G$ is a connected graph with the unit Perron vector $\mathbf{x}$, then any independent set $S$ of $G$ satisfies $\sum_{v\in S} x_v^2 \le \frac{1}{2}$, with equality if and only if $G$ is a bipartite graph and $S$ is one of the partite sets. Let $χ(G)= k $ be the chromatic number of $G$. A well-known conjecture of Gregory asserts that any independent set $S$ of $G$ satisfies $\frac{1}{2} - \sum_{v\in S}x_v^2 = Ω((k/n)^{1/2})$. Recently, Liu and Ning [J. Combin. Theory Ser. B 176 (2026)] disproved Gregory's conjecture by constructing a graph $G$ and an independent set $S$ such that $\frac{1}{2}- \sum_{v\in S}x_v^2 = O(k^5/n^3)$. Furthermore, they conjectured that this bound is tight up to a constant factor. In this paper, we first show that any cycle $C_n$ with odd integer $n\ge 7$ provides a simple counterexample to Gregory's conjecture. Second, we establish that for any independent set $S$, we have $\frac{1}{2} - \sum_{v\in S}x_v^2 = \frac{q}{4λ-2q}$, where $λ$ is the spectral radius of $G$, and $q$ is the Rayleigh quotient of $\mathbf{x}$ restricted to $\bar{S} :=V(G)\setminus S$. Third, we construct a graph with arbitrarily large chromatic number and find an independent set $S$ such that $\sum_{v\in S}x_v^2$ can be arbitrarily close to $\frac{1}{2}$, with an exponentially small gap. Our construction shows that there is no universal lower bound of the form $Ω(k^α/n^β)$ for any $α, β>0$. This settles both Gregory's original conjecture and the modified conjecture of Liu and Ning in the negative. Finally, we show the tightness of our construction and provide some local weighted lower bounds.

math.CO↗

arXiv2026

On Scott's odd induced subgraph conjecture and a related problem

For a graph $G$, let $f_o(G)$ denote the maximum order of an induced subgraph of $G$ all of whose vertices have odd degree, and let $χ(G)$ denote the chromatic number of $G$. Scott (CPC, 1992) proved that $f_o(G) \ge |V(G)|/(2χ(G))$ for every graph without isolated vertices, and conjectured that the factor $2$ can be removed. Wang and Wu (JGT, 2024) showed that this conjecture fails for bipartite graphs, but holds for line graphs. In this article, we confirm Scott's conjecture for claw-free graphs without isolated vertices, thereby strengthening the result of Wang and Wu. We also construct $K_{1,r}$-free graphs of arbitrarily large order to show that the conjecture fails for this broader class, for every integer $r \ge 4$. Wang and Wu also asked whether $f_o(L(G)) \ge n/2$ holds for every connected regular graph $G$ of order $n \ge 3$. We show that $C_5$ is the smallest counterexample to this problem. On the positive side, we prove that if $G$ is a connected $k$-regular $C_5$-free graph on $n$ vertices with $k \ge 2$, then $f_o(L(G)) \ge n/2$.

math.CO↗

arXiv2026

On derivatives and higher-order derivatives of chromatic polynomials

Let $ G $ be a graph of order $ n $ with maximum degree $Δ$, and let $P(G,x)$ denote its chromatic polynomial. We investigate several properties of $P(G,x)$ related to its derivatives and higher-order derivatives. First, we study the monotonicity of $P(G,x)/x^n$. Dong proved that $(x-1)^nP(G,x)\geq x^nP(G,x-1)$ for all real $x\geq n$. In particular, taking $x=n$ establishes the Bartels-Welsh ``shameful conjecture" that $P(G,n)/P(G,n-1)>e$. Fadnavis later showed that the same inequality holds for all real $x\geq 36Δ^{3/2}$. We improve this bound by proving that it also holds for all real $x\geq 10Δ^{3/2}$. We then consider a conjecture of Dong, Ge, Gong, Ning, Ouyang, and Tay asserting that $ \frac{d^k}{dx^k} \bigl( \ln[(-1)^n P(G, x)] \bigr) < 0 $ for all $ k \geq 2 $ and $ x \in (-\infty, 0) $. We establish this conjecture for all $ k \geq 2 $ and $ x\leq -3.01Δk $.

math.CO↗

arXiv2026

A short proof of a perturbation inequality for the spectral radius

Let $G$ be a simple graph, and denote by $λ(G)$ its spectral radius. Sun and Das (2020) established that for any non-isolated vertex $v$ with degree $d(v)$, \[ λ(G)\leq \sqrt{λ(G-v)^2 + 2d(v) - 1}, \] which is a conjecture original posed by Guo, Wang, and Li (2019). Sun and Das's proof uses several tools from spectral graph theory. In this short note, we provide a concise and self-contained proof of this inequality using matrix analysis.

math.CO↗

arXiv2026

An Improved Interpolation Theorem and Disproofs of Two Conjectures on 2-Connected Subgraphs

We prove that any $2$-connected graph $G$ on $n$ vertices with minimum degree $δ(G) \ge \frac{n}{4}+2$ contains a $2$-connected subgraph of order $k$ for every integer $k$ with $4 \le k \le n$. This improves a previous result of Yin and Wu. In \cite{YinWu-DAM-2026}, Yin and Wu proposed two conjectures. The first states that for any $2$-connected graph $G$ of order $n$ and size $m$, there exists a $2$-connected subgraph of order $k$ for each $k \in \{4, \dots, n\}$ whenever $m \ge \frac{1}{2} n^{3/2}$. The second conjecture asserts that the same conclusion holds under the alternative condition $δ(G) \ge \sqrt{n}$. In this paper, we construct counterexamples that completely disprove the first conjecture. Furthermore, using the existence of $(v, k, 2)$-Symmetric Balanced Incomplete Block designs (i.e., SBIBDs), we disprove the second conjecture for all $n \in \{8, 14, 22, 32, 74, 112, 158\}$. Finally, we propose a conjecture of our own: for any $2$-connected graph $G$ on $n$ vertices with $δ(G) \ge \frac{n}{k}$, where $k \ge 3$ and $n$ is sufficiently large, $G$ contains a $2$-connected subgraph of every order from $4$ to $n$.

math.CO↗

arXiv2025

A new spectral Turán theorem for weighted graphs and consequences

Confirming a conjecture of Elphick and Edwards and strengthening a spectral theorem of Wilf, Nikiforov proved that for any $K_{r+1}$-free graph $G$, $λ(G)^2 \leq 2 (1 - 1/r) m$, where $λ(G)$ is the spectral radius of $G$, and $m$ is the number of edges of $G$. This result was later improved in \cite{LiuN26}, where it was shown that for any graph $G$, $λ(G)^2 \leq 2 \sum_{e \in E(G)} \frac{\mathrm{cl}(e) - 1}{\mathrm{cl}(e)}$, where $\mathrm{cl}(e)$ denotes the order of the largest clique containing the edge $e$. In this paper, we further extend this inequality to weighted graphs, proving that \[ λ(G)^2 \leq 2 \sum_{e \in E(G)} \frac{\mathrm{cl}(e) - 1}{\mathrm{cl}(e)} w(e)^2, \] and we characterize all extremal graphs attaining this bound. Our main theorem yields several new consequences, including two vertex-based and vertex-degree-based local versions of Turán's theorem, as well as weighted generalizations of the Edwards--Elphick theorem and the Cvetković theorem, and two localized versions of Wilf's theorems. One of these localized Wilf's theorem confirms a conjecture that originates from Probability and Operator Algebras and was proposed by R. Tripathi independently of us. Moreover, our main result unifies and implies numerous earlier ones from spectral graph theory and extremal graph theory, including Stanley's spectral inequality, Hong's inequality, a localized Turán-type theorem, and a recent extremal theorem by Adak and Chandran. Notably, while Nikiforov's earlier spectral inequality implied Stanley's bound, it did not imply Hong's inequality -- a gap that is now bridged by our result. As a key tool, we establish the inequality $\sum_{e \in E(G)} \frac{2}{\mathrm{cl}(e)} \geq n-1$, which complements an upper bound $\sum_{e \in E(G)} \frac{2}{\mathrm{cl}(e)-1} \leq n^2 - 2m$ due to Bradač, and Malec and Tompkins, independently.

math.CO↗

arXiv2025

Exact Turán numbers of two vertex-disjoint paths

The Turán number of a graph $H$ is the maximum number of edges in any graph of order $n$ that does not contain $H$ as a subgraph. In 1959, Erd\H os and Gallai obtained a sharp upper bound of Turán numbers for a path of arbitrary length. In 1975, Faudree and Schelp, and independently in 1977, Kopylov determined the exact values of Turán numbers of paths with arbitrary length. In this paper, we determine the Turán number of two vertex-disjoint paths of odd order at least 4. Together with previous works, we determine the exact Turán numbers of two vertex-disjoint paths completely. This confirms the first $k=2$ case of a conjecture proposed by Yuan and Zhang in 2021, which generalizes the Turán number formula of paths due to Faudree-Schelp, and Kopylov in a broader setting. Our main tools include a refinement of Pósa's rotation lemma, a stability result of Kopylov's theorem on cycles, and a recent inequality on circumference, minimum degree, and clique number of a 2-connected graph.

math.CO↗

arXiv2025

Two conjectures on vertex-disjoint rainbow triangles

In 1963, Dirac proved that every $n$-vertex graph has $k$ vertex-disjoint triangles if $n\geq 3k$ and minimum degree $δ(G)\geq \frac{n+k}{2}$. The base case $n=3k$ can be reduced to the Corrádi-Hajnál Theorem. Towards a rainbow version of Dirac's Theorem, Hu, Li, and Yang conjectured that for all positive integers $n$ and $k$ with $n\geq 3k$, every edge-colored graph $G$ of order $n$ with $δ^c(G)\geq \frac{n+k}{2}$ contains $k$ vertex-disjoint rainbow triangles. In another direction, Wu et al. conjectured an exact formula for anti-Ramsey number $ar(n,kC_3)$, generalizing the earlier work of Erdős, Sós and Simonovits. The conjecture of Hu, Li, and Yang was confirmed for the cases $k=1$ and $k=2$. However, Lo and Williams disproved the conjecture when $n\leq \frac{17k}{5}.$ It is therefore natural to ask whether the conjecture holds for $n=Ω(k)$. In this paper, we confirm this by showing that the Hu-Li-Yang conjecture holds when $n\ge 42.5k+48$. We disprove the conjecture of Wu et al. and propose a modified conjecture. This conjecture is motivated by previous works due to Allen, Böttcher, Hladký, and Piguet on Turán number of vertex-disjoint triangles.

math.CO↗

arXiv2025

Localized and weighted versions of extremal problems

Malec and Tompkins (EUJC, 2023) considered the localized versions of Turán-type problems, and proved a localized theorem on Erdős-Gallai Theorem on paths. Zhao and Zhang (JGT, 2025) gave a long proof of a localized version of Erdős-Gallai Theorem on cycles. In this paper, we consider several types of generalization of Turán-type problems, that is, localized versions, weighted versions, and generalized Turán-type problems, and their connectedness. We first present very short proofs for recent results of Malec-Tompkins and Zhao-Zhang, respectively. We use Small Path Double Cover Conjecture, which was proposed by Bondy (JGT, 1990) and confirmed by Hao Li (JGT, 1990), to prove a weighted localized Turán-type theorem on paths. We prove localized versions of Balister-Bollobás-Riordan-Schelp Theorem (JCTB, 2003) on paths and Erdős-Gallai Theorem on matchings, respectively. We show that our first localized result implies Balister-Bollobás- Riordan-Schelp Theorem, Erdős-Gallai Theorem, and Malec-Tompkins Theorem on paths. Finally, we present generalized Turán-style generalizations of the Malec-Tompkin's Theorem, and discuss the relationship between some previous theorems in different motivations.

math.CO↗

arXiv2025

Efficient Size Constraint Community Search over Heterogeneous Information Networks

The goal of community search in heterogeneous information networks (HINs) is to identify a set of closely related target nodes that includes a query target node. In practice, a size constraint is often imposed due to limited resources, which has been overlooked by most existing HIN community search works. In this paper, we introduce the size-bounded community search problem to HIN data. Specifically, we propose a refined (k, P)-truss model to measure community cohesiveness, aiming to identify the most cohesive community of size s that contains the query node. We prove that this problem is NP-hard. To solve this problem, we develop a novel B\&B framework that efficiently generates target node sets of size s. We then tailor novel bounding, branching, total ordering, and candidate reduction optimisations, which enable the framework to efficiently lead to an optimum result. We also design a heuristic algorithm leveraging structural properties of HINs to efficiently obtain a high-quality initial solution, which serves as a global lower bound to further enhance the above optimisations. Building upon these, we propose two exact algorithms that enumerate combinations of edges and nodes, respectively. Extensive experiments on real-world datasets demonstrate the effectiveness and efficiency of the proposed methods.

cs.DB↗

arXiv2025

The stability of independence polynomials of complete bipartite graphs

The independence polynomial of a graph is termed {\it stable} if all its roots are located in the left half-plane $\{z \in \mathbb{C} : \mathrm{Re}(z) \leq 0\}$, and the graph itself is also referred to as stable. Brown and Cameron (Electron. J. Combin. 25(1) (2018) \#P1.46) proved that the complete bipartite graph $K_{1,n}$ is stable and posed the question: \textbf{Are all complete bipartite graphs stable?} We answer this question by establishing the following results: \begin{itemize} \item The complete bipartite graphs $K_{2,n}$ and $K_{3,n}$ are stable. \item For any integer $k\geq0$, there exists an integer $N(k)\in \mathbb{N}$ such that $K_{m,m+k}$ is stable for all $m>N(k)$. \item For any rational $\ell> 1$, there exists an integer $N(\ell) \in \mathbb{N}$ such that whenever $m >N(\ell)$ and $\ell \cdot m$ is an integer, $K_{m, \ell \cdot m}$ is \textbf{not} stable. \end{itemize}

math.CO↗

arXiv2024

The number of edges in graphs with bounded clique number and circumference

Let $\cal H$ be a family of graphs. The Turán number ${\rm ex}(n,{\cal H})$ is the maximum possible number of edges in an $n$-vertex graph which does not contain any member of $\cal H$ as a subgraph. As a common generalization of Turán's theorem and Erdős-Gallai theorem on the Turán number of matchings, Alon and Frankl determined ${\rm ex}(n,{\cal H})$ for ${\cal H}=\{K_r,M_k\}$, where $M_k$ is a matching of size $k$. Replacing $M_k$ by $P_k$, Katona and Xiao obtained the Turán number of ${\cal H}=\{K_r,P_k\}$ for $r \leq \lfloor k/2 \rfloor$ and sufficiently large $n$. In addition, they proposed a conjecture for the case of $r \geq \lfloor k/2 \rfloor+1$ and sufficiently large $n$. Motivated by the fact that the result for ${\rm ex}(n,P_k)$ can be deduced from the one for ${\rm ex}(n,{\cal C}_{\geq k})$, we investigate the Turán number of ${\cal H}=\{K_r, {\cal C}_{\geq k}\}$ in this paper. In other words, we aim to determine the maximum number of edges in graphs with clique number at most $r-1$ and circumference at most $k-1$. For ${\cal H}=\{K_r, {\cal C}_{\geq k}\}$, we are able to show the value of ${\rm ex}(n,{\cal H})$ for $r \geq \lfloor (k-1)/2\rfloor+2$ and all $n$. As an application of this result, we confirm Katona and Xiao's conjecture in a stronger form. For $r \leq \lfloor (k-1)/2\rfloor+1$, we manage to show the value of ${\rm ex}(n,{\cal H})$ for sufficiently large $n$.

math.CO↗

arXiv2024

On the two problems in Ramsey achievement games

Let $p,q$ be two integers with $p\geq q$. Given a finite graph $F$ with no isolated vertices, the generalized Ramsey achievement game of $F$ on the complete graph $K_n$, denoted by $(p,q;K_n,F,+)$, is played by two players called Alice and Bob. In each round, Alice firstly chooses $p$ uncolored edges $e_1,e_2,...,e_p$ and colors it blue, then Bob chooses $q$ uncolored edge $f_1,f_2,...,f_q$ and colors it red; the player who can first complete the formation of $F$ in his (or her) color is the winner. The generalized achievement number of $F$, denoted by ${a}(p,q;F)$ is defined to be the smallest $n$ for which Alice has a winning strategy. If $p=q=1$, then it is denoted by ${a}(F)$, which is the classical achievement number of $F$ introduced by Harary in 1982. If Alice aims to form a blue $F$, and the goal of Bob is to try to stop him, this kind of game is called the first player game by Bollobás. Let ${a}^*(F)$ be the smallest positive integer $n$ for which Alice has a winning strategy in the first player game. A conjecture due to Harary states that the minimum value of ${a}(T)$ is realized when $T$ is a path and the maximum value of ${a}(T)$ is realized when $T$ is a star among all trees $T$ of order $n$. He also asked which graphs $F$ satisfy $a^*(F)=a(F)$? In this paper, we proved that $n\leq {a}(p,q;T)\leq n+q\left\lfloor (n-2)/p \right\rfloor$ for all trees $T$ of order $n$, and obtained a lower bound of ${a}(p,q;K_{1,n-1})$, where $K_{1,n-1}$ is a star. We proved that the minimum value of ${a}(T)$ is realized when $T$ is a path which gives a positive solution to the first part of Harary's conjecture, and ${a}(T)\leq 2n-2$ for all trees of order $n$. We also proved that for $n\geq 3$, we have $2n-2-\sqrt{(4n-8)\ln (4n-4)}\leq a(K_{1,n-1})\leq 2n-2$ with the help of a theorem of Alon, Krivelevich, Spencer and Szabó. We proved that $a^*(P_n)=a(P_n)$ for a path $P_n$.

math.CO↗

arXiv2024

The generalized Tur'{a}n number of long cycles in graphs and bipartite graphs

Given a graph $T$ and a family of graphs $\mathcal{F}$, the maximum number of copies of $T$ in an $\mathcal{F}$-free graph on $n$ vertices is called the generalized Turán number, denoted by $ex(n, T , \mathcal{F})$. When $T= K_2$, it reduces to the classical Turán number $ex(n, \mathcal{F})$. Let $ex_{bip}(b,n, T , \mathcal{F})$ be the maximum number of copies of $T$ in an $\mathcal{F}$-free bipartite graph with two parts of sizes $b$ and $n$, respectively. Let $P_k$ be the path on $k$ vertices, $\mathcal{C}_{\ge k}$ be the family of all cycles with length at least $k$ and $M_k$ be a matching with $k$ edges. In this article, we determine $ex_{bip}(b,n, K_{s,t}, \mathcal{C}_{\ge 2n-2k})$ exactly in a connected bipartite graph $G$ with minimum degree $δ(G) \geq r\ge 1$, for $b\ge n\ge 2k+2r$ and $k\in \mathbb{Z}$, which generalizes a theorem of Moon and Moser, a theorem of Jackson and gives an affirmative evidence supporting a conjecture of Adamus and Adamus. As corollaries of our main result, we determine $ex_{bip}(b,n, K_{s,t}, P_{2n-2k})$ and $ex_{bip}(b,n, K_{s,t}, M_{n-k})$ exactly in a connected bipartite graph $G$ with minimum degree $δ(G) \geq r\ge 1$, which generalizes a theorem of Wang. Moreover, we determine $ex(n, K_{s,t}, \mathcal{C}_{\ge k})$ and $ex(n, K_{s,t}, P_{k})$ respectively in a connected graph $G$ with minimum degree $δ(G) \geq r\ge 1$, which generalizes a theorem of Lu, Yuan and Zhang.

math.CO↗

arXiv2024

Violation of Weak Cosmic Censorship in de Sitter Space

Inspired by the recent discovery of a violation of strong cosmic censorship (SCC) for the near-extremal Reissner-Nordström black holes in de Sitter space (RN-dS), we investigate if the weak cosmic censorship conjecture (WCCC) can also be violated in RN-dS with a fixed cosmological constant. Our method is based on the recent formulation of examining WCCC by requiring the second law to hold, which requires the sum of areas of the event and cosmic horizons cannot decrease during the infall process of Wald's gedanken experiment. We find that the WCCC can be violated for the near-extremal RN-dS in some regimes of second-order perturbation of field configurations. Given the charge parameter of RN-dS, we can find the lowest value of the sub-extremality parameter, beyond which the WCCC holds. Our results imply that violations of SCC and WCCC could be correlated. Because of a lack of an unambiguous relation between the gravitational mass and matter's kinematic mass in asymptotically de Sitter space, we cannot compare the corresponding regimes of parameter space for the violations of SCC and WCCC. We also discuss the subtlety in formulating both the first-law and second-law approaches to examine WCCC.

hep-th↗

arXiv2024

On degree power sum in $P_k$-free graphs

Let $G$ be a graph on $n$ vertices with degree sequence $(d_1,d_2......d_n)$. For a real $p \geq 1$, let $D_p(G)=\sum_{i=1}^nd_i^p$. A Turán-type problem of degree power sum was initiated by Caro and Yuster \cite{caro2000degpower}: determining the function $D_p(n,H) :=\max \{D_p(G): \text{$G$ is an $n$-vertex $H$-free graph}\}$. They obtained some exact values for certain graphs $H$. For a path $P_k$, they mentioned that ``a close examination of the proof of Theorem 1.2 shows that the value of $n_0(k)$ in the statement of the theorem is $O(k^2)$", namely, they could show the $n$-vertex $P_k$-free graph with maximum degree power sum is $W_{n,k-1,\lfloor \frac{k}{2} \rfloor -1} = K_{\lfloor \frac{k}{2} \rfloor -1} \vee \left((n - \lceil \frac{k}{2} \rceil)K_1 \cup K_{1+k-2\lfloor \frac{k}{2} \rfloor} \right)$ when $n \geq c k^2$ for some constant $c$. In this note, we improve their result to a linear size of $k$ by a different approach. The bound is tight up to a constant factor.

math.CO↗

arXiv2024

Variants of spectral Turán theorems and eigenvectors of graphs

In 2002, Nikiforov proved that for an $n$-vertex graph $G$ with clique number $ω$ and edge number $m$, the spectral radius $λ(G)$ satisfies $λ(G) \leq \sqrt{2(1 - 1/ω) m}$, which confirmed a conjecture implicitly suggested by Edwards and Elphick. In this paper, we prove a local version of spectral Turán inequality, which states that $λ^2(G)\leq 2\sum_{e\in E(G)}\frac{c(e)-1}{c(e)}$, where $c(e)$ is the order of the largest clique containing the edge $e$ in $G$. We also characterize the extremal graphs. We prove that our theorem implies Nikiforov's theorem and give an example to show that the difference of Nikiforov's bound and ours is $Ω(\sqrt{m})$ for some cases. Additionally, we establish a spectral counterpart to Ore's problem (1962) which asks for the maximum size of an $n$-vertex graph such that its complement is connected and does not contain $F$ as a subgraph. Our result leads to a new spectral Turán inequality applicable to graphs with connected complements. Finally, we disprove a conjecture of Gregory, asserting that for a connected $n$-vertex graph $G$ with chromatic number $k\geq 2$ and an independent set $S$, we have \[ \sum_{v\in S} x_v^2 \leq \frac{1}{2} - \frac{k-2}{2\sqrt{(k-2)^2 + 4(k-1)(n-k+1)}}, \] where $x_v$ is the component of the Perron vector of $G$ with respect to the vertex $v$. A modified version of Gregory's conjecture is proposed.

math.CO↗