arXiv subjects
Publications and source records attributed to Lele Liu.
Boots and Royle, and independently Cao and Vince, conjectured that the join of an edge and a path on $n-2$ vertices is the unique planar graph of maximum adjacency spectral radius for $n\geq 9$. Tait and Tobin (JCTB, 2017) proved the conjecture for sufficiently large order. In this paper, we completely resolved the Boots-Royle/Cao-Vince conjecture.
Let $G$ be a simple graph of maximum degree $d$, and let $\mu(G)$ denote the largest eigenvalue of its Laplacian matrix. For a fixed integer $k\geq 2$, Aharoni, Alon, and Berger (2016) asked whether every graph containing no induced copy of $K_{1,k}$ satisfies $\mu(G)\leq (2 - \frac{2}{k} + o(1)) d$. We answer this question by proving the stronger sharp bound \[ \mu(G)\leq \left(2-\frac{2}{k}\right)(d+1). \] The proof combines a sign decomposition of a Laplacian Rayleigh vector with a weighted local Caro-Wei type inequality for independent sets.
Let $G$ be an $n$-vertex graph, and let $\lambda(G)$ and $\lambda_n(G)$ 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 Tur\'an graph on $n$ vertices. We prove four sharp results in spectral Tur\'an theory. First, we confirm Guiduli's spectral dense-neighborhood conjecture (1996) in a stronger form: if $\lambda(G)\ge \lambda(T_r(n))$, then either $G\cong T_r(n)$, or there exists a vertex $v$ such that $\lambda(G[N(v)]) > \lambda(T_{r-1}(d(v)))$. Moreover, when $\lambda(G)>\lambda(T_r(n))$, 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 Tur\'an edge threshold is detected by the exact spectral threshold: for every $r\ge 2$ and every $n$, $\lambda(G)<\lambda(T_r(n))$, 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 \[ \omega(G)\ge 1+\frac{2e(G)}{(n-d(G))(d(G)-\lambda_n(G))} \] does imply the concise form of Tur\'an's theorem. Finally, we discuss an open problem proposed by Ai et al. (2026) in \cite{ALNS26+}.
Wang, Cui, and Cioab\u{a} introduced the Laplacian matching root integral variation of a graph and proved that it cannot occur in one place. They also showed that the two-place variation is impossible for connected graphs satisfying $g(G)/c(G)>7/6$, where $g(G)$ is the girth and $c(G)$ is the dimension of the cycle space, and conjectured that no connected graph admits such a two-place variation. In this paper, we confirm this conjecture. The proof combines a structural relation obtained in their paper with two new power-sum identities for Laplacian matching roots.
A classical result of Cioab\u{a} 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 $\chi(G)= k $ be the chromatic number of $G$. A well-known conjecture of Gregory asserts that any independent set $S$ of $G$ satisfies $\frac{1}{2} - \sum_{v\in S}x_v^2 = \Omega ((k/n)^{1/2})$. 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 $\frac{1}{2} - \sum_{v\in S}x_v^2 = \frac{q}{4\lambda -2q}$, where $\lambda$ 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 $\Omega (k^{\alpha}/n^{\beta})$ for any $\alpha, \beta >0$. 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.
For a graph $G$ of order $n$, the spectral sum of $G$ is defined to be the sum $\lambda_1(G) + \lambda_2(G)$, where $\lambda_1(G)$ (resp. $\lambda_2(G)$) is the largest (resp. second largest) adjacency eigenvalue of $G$. Ebrahimi, Mohar, Nikiforov and Ahmady (2008) conjectured that the spectral sum \[ \lambda_1(G) + \lambda_2(G)\le \frac{8}{7}n \] for any graph $G$. We prove this conjecture by combining tools from the theory of graph limits, convex geometry, exterior algebra and convex optimization. The techniques developed are of independent interest.
Let $G$ be a simple graph, and denote by $\lambda(G)$ its spectral radius. Sun and Das (2020) established that for any non-isolated vertex $v$ with degree $d(v)$, \[ \lambda(G)\leq \sqrt{\lambda(G-v)^2 + 2d(v) - 1}, \] 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.
Confirming a conjecture of Elphick and Edwards and strengthening a spectral theorem of Wilf, Nikiforov proved that for any $K_{r+1}$-free graph $G$, $\lambda(G)^2 \leq 2 (1 - 1/r) m$, where $\lambda(G)$ 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 $G$, $\lambda(G)^2 \leq 2 \sum_{e \in E(G)} \frac{\mathrm{cl}(e) - 1}{\mathrm{cl}(e)}$, 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 \[ \lambda(G)^2 \leq 2 \sum_{e \in E(G)} \frac{\mathrm{cl}(e) - 1}{\mathrm{cl}(e)} w(e)^2, \] 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 Tur\'an's theorem, as well as weighted generalizations of the Edwards--Elphick theorem and the Cvetkovi\'c 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 Tur\'an-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 Brada\v{c}, and Malec and Tompkins, independently.
The ongoing energy crisis has underscored the urgent need for energy-efficient materials with high energy utilization efficiency, prompting a surge in research into organic compounds due to their environmental compatibility, cost-effective processing, and versatile modifiability. To address the high experimental costs and time-consuming nature of traditional trial-and-error methods in the discovery of highly functional organic compounds, we apply the 3D transformer-based molecular representation learning algorithm to construct a pre-trained model using 60 million semi-empirically optimized structures of small organic molecules, namely, Org-Mol, which is then fine-tuned with public experimental data to obtain prediction models for various physical properties. Despite the pre-training process relying solely on single molecular coordinates, the fine-tuned models achieves high accuracy (with $R^2$ values for the test set exceeding 0.95). These fine-tuned models are applied in a high-throughput screening process to identify novel immersion coolants among millions of automatically constructed ester molecules, resulting in the experimental validation of two promising candidates. This work not only demonstrates the potential of Org-Mol in predicting bulk properties for organic compounds but also paves the way for the rational and efficient development of ideal candidates for energy-saving materials.
The spread of a real symmetric matrix is defined as the difference between its largest and smallest eigenvalue. The study of graph-related matrices has attracted considerable attention, leading to a substantial body of findings. In this paper, we investigate a general spread problem related to $A_{\alpha}$-matrix of graphs. The $A_{\alpha}$-matrix of a graph $G$, introduced by Nikiforov in 2017, is a convex combinations of its diagonal degree matrix $D(G)$ and adjacency matrix $A(G)$, defined as $A_{\alpha} (G) = \alpha D(G) + (1-\alpha) A(G)$. Let $\lambda_1^{(\alpha)} (G)$ and $\lambda_n^{(\alpha)} (G)$ denote the largest and smallest eigenvalues of $A_{\alpha} (G)$, respectively. We determined the unique graph that maximizes $\lambda^{(\alpha)}_1 (G) - \beta\cdot\lambda^{(\gamma)}_n (G)$ among all connected $n$-vertex graphs for sufficiently large $n$, where $0 \leq \alpha < 1$, $1/2\leq \gamma < 1$ and $0<\beta\gamma\leq 1$. As an application, we confirm a conjecture proposed by Lin, Miao, and Guo [Linear Algebra Appl. 606 (2020) 1--22]. In addition, one of main results in [SIAM J. Discrete Math. 38 (2024) 590--608] is a simple corollary of our result by choosing $\alpha = \gamma = 1/2$ and $\beta = 1$.
The spread of a graph is the difference between the largest and smallest eigenvalue of its adjacency matrix. In this paper, we investigate spread problems for graphs with excluded clique-minors. We show that for sufficiently large $n$, the $n$-vertex $K_r$-minor free graph with maximum spread is the join of a clique and an independent set, with $r-2$ and $n-r+2$ vertices, respectively.
The spectral Tur\'an theorem states that the $k$-partite Tur\'an graph is the unique graph attaining the maximum adjacency spectral radius among all graphs of order $n$ containing no the complete graph $K_{k+1}$ as a subgraph. This result is known to be stronger than the classical Tur\'an theorem. In this paper, we consider hypergraph extensions of spectral Tur\'an theorem. For $k\geq r\geq 2$, let $H_{k+1}^{(r)}$ be the $r$-uniform hypergraph obtained from $K_{k+1}$ by enlarging each edge with a new set of $(r-2)$ vertices. Let $F_{k+1}^{(r)}$ be the $r$-uniform hypergraph with edges: $\{1,2,\ldots,r\} =: [r]$ and $E_{ij} \cup\{i,j\}$ over all pairs $\{i,j\}\in \binom{[k+1]}{2}\setminus\binom{[r]}{2}$, where $E_{ij}$ are pairwise disjoint $(r-2)$-sets disjoint from $[k+1]$. Generalizing the Tur\'an theorem to hypergraphs, Pikhurko [J. Combin. Theory Ser. B, 103 (2013) 220--225] and Mubayi and Pikhurko [J. Combin. Theory Ser. B, 97 (2007) 669--678] respectively determined the exact Tur\'an number of $H_{k+1}^{(r)}$ and $F_{k+1}^{(r)}$, and characterized the corresponding extremal hypergraphs. Our main results show that $T_r(n,k)$, the complete $k$-partite $r$-uniform hypergraph on $n$ vertices where no two parts differ by more than one in size, is the unique hypergraph having the maximum $p$-spectral radius among all $n$-vertex $H_{k+1}^{(r)}$-free (resp. $F_{k+1}^{(r)}$-free) $r$-uniform hypergraphs for sufficiently large $n$. These findings are obtained by establishing $p$-spectral version of the stability theorems. Our results offer $p$-spectral analogues of the results by Mubayi and Pikhurko, and connect both hypergraph Tur\'an theorem and hypergraph spectral Tur\'an theorem in a unified form via the $p$-spectral radius.
Let $G$ be a simple connected graph of order $n$ and $\partial(G)$ is the spectral radius of the distance matrix $D(G)$ of $G$. The transmission $D_i$ of vertex $i$ is the $i$-th row sum of $D(G)$. Denote by $D_{\max}(G)$ the maximum of transmissions over all vertices of $G$, and $\partial^Q(G)$ is the spectral radius of the distance signless Laplacian matrix $D(G)+\mbox{diag}(D_1,D_2,\ldots,D_n)$. In this paper, we present a sharp lower bound of $2D_{\max}(G)-\partial^Q(G)$ among all $n$-vertex connected graphs, and characterize the extremal graphs. Furthermore, we give the minimum values of respective $D_{\max}(G)-\partial(G)$ and $2D_{\max}(G)-\partial^Q(G)$ on trees and characterize the extremal trees.
In 2002, Nikiforov proved that for an $n$-vertex graph $G$ with clique number $\omega$ and edge number $m$, the spectral radius $\lambda(G)$ satisfies $\lambda (G) \leq \sqrt{2(1 - 1/\omega) m}$, which confirmed a conjecture implicitly suggested by Edwards and Elphick. In this paper, we prove a local version of spectral Tur\'an inequality, which states that $\lambda^2(G)\leq 2\sum_{e\in E(G)}\frac{c(e)-1}{c(e)}$, 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 $\Omega (\sqrt{m})$ 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 Tur\'an 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.
Artificial Intelligence (AI) technologies such as deep learning are evolving very quickly bringing many changes to our everyday lives. To explore the future impact and potential of AI in the field of music and sound technologies a doctoral day was held between Queen Mary University of London (QMUL, UK) and Sciences et Technologies de la Musique et du Son (STMS, France). Prompt questions about current trends in AI and music were generated by academics from QMUL and STMS. Students from the two institutions then debated these questions. This report presents a summary of the student debates on the topics of: Data, Impact, and the Environment; Responsible Innovation and Creative Practice; Creativity and Bias; and From Tools to the Singularity. The students represent the future generation of AI and music researchers. The academics represent the incumbent establishment. The student debates reported here capture visions, dreams, concerns, uncertainties, and contentious issues for the future of AI and music as the establishment is rightfully challenged by the next generation.
Let $F$ be a graph and $\SPEX (n, F)$ be the class of $n$-vertex graphs which attain the maximum spectral radius and contain no $F$ as a subgraph. Let $\EX (n, F)$ be the family of $n$-vertex graphs which contain maximum number of edges and no $F$ as a subgraph. It is a fundamental problem in spectral extremal graph theory to characterize all graphs $F$ such that $\SPEX (n, F)\subseteq \EX (n, F)$ when $n$ is sufficiently large. Establishing the conjecture of Cioab\u{a}, Desai and Tait [European J. Combin., 2022], Wang, Kang, and Xue [J. Combin. Theory Ser. B, 2023] prove that: for any graph $F$ such that the graphs in $\EX (n, F)$ are Tur\'{a}n graphs plus $O(1)$ edges, $\SPEX (n, F)\subseteq \EX (n, F)$ for sufficiently large $n$. In this paper, we prove that $\SPEX (n, F)\subseteq \EX (n, F)$ for sufficiently large $n$, where $F$ is an $n$-vertex graph with no isolated vertices and $\Delta (F) \leq \sqrt{n}/40$. We also prove a signless Laplacian spectral radius version of the above theorem. These results give new contribution to the open problem mentioned above, and can be seen as spectral analogs of a theorem of Alon and Yuster [J. Combin. Theory Ser. B, 2013]. Furthermore, as immediate corollaries, we have tight spectral conditions for the existence of several classes of special graphs, including clique-factors, $k$-th power of Hamilton cycles and $k$-factors in graphs. The first special class of graphs gives a positive answer to a problem of Feng, and the second one extends a previous result of Yan et al.
Spectral graph theory is a captivating area of graph theory that employs the eigenvalues and eigenvectors of matrices associated with graphs to study them. In this paper, we present a collection of $20$ topics in spectral graph theory, covering a range of open problems and conjectures. Our focus is primarily on the adjacency matrix of graphs, and for each topic, we provide a brief historical overview.
Let $G$ be a cancellative $3$-uniform hypergraph in which the symmetric difference of any two edges is not contained in a third one. Equivalently, a $3$-uniform hypergraph $G$ is cancellative if and only if $G$ is $\{F_4, F_5\}$-free, where $F_4 = \{abc, abd, bcd\}$ and $F_5 = \{abc, abd, cde\}$. A classical result in extremal combinatorics stated that the maximum size of a cancellative hypergraph is achieved by the balanced complete tripartite $3$-uniform hypergraph, which was firstly proved by Bollobás and later by Keevash and Mubayi. In this paper, we consider spectral extremal problems for cancellative hypergraphs. More precisely, we determine the maximum $p$-spectral radius of cancellative $3$-uniform hypergraphs, and characterize the extremal hypergraph. As a by-product, we give an alternative proof of Bollobás' result from spectral viewpoint.