arXiv subjects
Publications and source records attributed to Xiaogang Liu.
In this paper, we establish two Turán-type results for signed graphs. We first generalize the classical Turán theorem to signed graphs and then extend Nikiforov's spectral Turán theorem to signed graphs. Moreover, we determine the second maximum spectral radius among all unbalanced signed graphs that contain no balanced complete signed subgraph on \(r+1\) vertices.
A persistent challenge in plasmonic upconversion is decoupling pump-field enhancement from emission-side local-density-of-optical-states (LDOS) engineering to achieve selective band manipulation. Here, we show that a corrugated SU8/Au/Al2O3 grating coated with a NaYF4:Yb/Er upconversion nanoparticle (UCNP) monolayer realizes a truly band-selective platform. A broad plasmonic resonance near 670 nm modulates the red Er 3 + 3+ decay rate by $\pm$15% as a function of the Al2O3 spacer thickness, while leaving the green transition experimentally invariant (< 1% change). Simultaneously, the 980 nm pump field is monotonically suppressed below free-space levels, ensuring that steady-state and time-resolved observables cleanly probe the emission-side LDOS without pump interference. We analyze this system using a coupled electromagnetic-kinetic framework that integrates finite-difference time-domain (FDTD) calculations of Purcell factors and pump fields with a six-level Yb/Er rate-equation model. The framework quantitatively reproduces the 670 nm plasmonic resonance, the red-band decay-rate modulation, and the monotonic decrease of the green/red intensity ratio. Crucially, the model serves as a powerful diagnostic tool: it overpredicts a green-band rate reduction, but systematic parametric testing rules out geometric (apex smoothing) and material (grain-boundary damping, interband loss) imperfections as the cause. Instead, it isolates the residual discrepancy to measurement-versus-model factors (finite-aperture angular averaging) and missing non-radiative kinetic channels, establishing a clear roadmap for the rational design and validation of future plasmonic-UCNP architectures.
Fractional revival (FR), a generalization of perfect state transfer (PST), is a significant phenomenon in quantum state transfer that allows quantum information to be transmitted between two qubits with a certain probability. The existence of FR has been extensively studied on many classes of graphs. However, oriented graphs have not yet been investigated. In this paper, we investigate the existence of proper FR on oriented graphs. We first establish necessary and sufficient conditions for oriented graphs to admit proper FR between strongly cospectral vertices. Furthermore, we prove that oriented Cayley graphs over abelian groups do not admit proper FR, and we subsequently characterize the conditions under which oriented semi-Cayley graphs over abelian groups admit proper FR.
Let $Γ= (G, σ)$ be a signed graph and $v$ a non-isolated vertex of $Γ$. Let $Γ-v$ denote the graph obtained by deleting the vertex $v$ together with all signed edges incident to it from $Γ$, and $d_Γ(v)$ the degree of $v$ in $Γ$. In this paper, we prove that the largest eigenvalue $λ_1(Γ)$ of $Γ$ satisfies \[ λ_1(Γ) \le \sqrt{λ_1^2(Γ- v) + 2d_Γ(v) - 1}, \] and we also present a refined version of this bound. Moreover, we characterize the extremal signed graphs achieving equality when $Γ$ is connected and $d_Γ(v)\ge 2$, which are switching equivalent to the balanced complete signed graph.
In this paper, we extend localized Turán theorems to signed graphs and study the corresponding spectral Turán problems. Firstly, we establish a vertex-localized Turán-type inequality for signed graphs and characterize the extremal graphs reaching this bound. Secondly, we derive a sharp upper bound for the largest eigenvalue of a signed graph via localized Turán parameters, and identify the extremal graphs attaining equality. Finally, we generalize a walk-based local spectral Turán inequality to signed graphs. Our results generalize and improve several existing theorems for unsigned and signed graphs.
A $k$-matching in a graph $G$ is defined as a function $f:E(G) \rightarrow \{0,1,\ldots,k\}$ satisfying $\sum_{e\in E_G(v)} f(e)$ $\leq k$ for each vertex $v\in V(G)$, where $E_G(v)$ denotes the set of edges incident to $v$ in $G$. For $1\leq d\leq k$ and $d \equiv |V(G)|~(\mathrm{mod}~2)$, if for any $ v \in V(G)$, there exists a $k$-matching $f$ such that $\sum_{e\in E_G(v)}f(e)=k-d$ and $\sum_{e\in E_G(u)}f(e)=k \text{ for any } u\in V(G)-\{v\}$, then $G$ is $k$-$d$-critical. A graph $G$ of odd order (resp. even order) is generalized factor-critical (resp. generalized bicritical) if the empty set is the unique set attaining the maximum value in $k$-Berge-Tutte-formula of $G$. In this paper, we provide sharp sufficient conditions in terms of size or spectral radius respectively for a graph $G$ to be $k$-$d$-critical, generalized factor-critical and generalized bicritical with minimum degree.
Let $tK_4$ denote the family of all graphs consisting of $t$ copies of $K_4$ that are allowed to share vertices and $t\mathcal{K}_{4}^{-}$ be the set of all unbalanced signed graphs whose underlying graphs are elements of $tK_4$. In this paper, we characterize the extremal graphs that achieve the maximum index and spectral radius among all $t\mathcal{K}_{4}^{-}$-free unbalanced signed graphs with given order.
In this paper, we consider the bounds for the largest eigenvalue and the sum of the $k$ largest Laplacian eigenvalues of signed graphs. Firstly, we give an upper bound on the largest eigenvalue of the adjacency matrix of a signed graph and characterize the extremal graphs that attain this bound. Secondly, we prove that a non-bipartite signed graph $Γ$ of order $n$ and size $m$ contains a balanced triangle if $λ_{1}(Γ)\ge \sqrt{m-1}$, $λ_{1}(Γ) \ge |λ_{n}(Γ)|$ and $Γ\not \sim (C_{5}\cup (n-5)K_{1},+)$, where $λ_{1}(Γ)$ is the largest eigenvalue of the adjacency matrix of $Γ$. Thirdly, we confirm a conjecture proposed in [Linear Multilinear Algebra 51 (1) (2003) 21--30] that: if $Γ$ is a connected signed graph, then $$ \sum_{i=1}^{k}μ_{i}(Γ) >\sum_{i=1}^{k}d_{i}(Γ)~~(1\le k\le n-1), $$ where $μ_{1}(Γ)\geμ_{2}(Γ)\ge\cdots \ge μ_{n}(Γ)$ are Laplacian eigenvalues of $Γ$, and $d_{1}(Γ)\ge d_{2}(Γ)\ge \dots \ge d_{n}(Γ)$ are vertex degrees of $Γ$. Finally, we give a lower bound for the sum of the $k$ largest Laplacian eigenvalues of a connected signed graph.
Quantum state transfer, first introduced by Bose in 2003, is an important physical phenomenon in quantum networks, which plays a vital role in quantum communication and quantum computing. In 2004, Christandl et al. proposed the concept of perfect state transfer on graphs by modeling the quantum network using graphs, and unveiled the feasibility of applying graph theory to quantum state transfer. In 2018, Chen and Godsil proposed the definition of Laplacian perfect pair state transfer on graphs, which is a brilliant generalization of perfect state transfer. In this paper, we investigate the existence of Laplacian perfect pair state transfer in tensor product and double cover of two regular graphs, respectively, and reveal fundamental connections between perfect state transfer and Laplacian perfect pair state transfer. We give necessary and sufficient conditions for the tensor product of two regular graphs to admit Laplacian perfect pair state transfer, where one of the two regular graphs admits perfect state transfer or Laplacian perfect pair state transfer. Additionally, we characterize the existence of Laplacian perfect pair state transfer in the double cover of two regular graphs. By our results, a variety of families of graphs admitting Laplacian perfect pair state transfer can be constructed.
Let $T_{8p} = \left\langle a,b\mid a^{2p}=b^8=e,a^p=b^4,b^{-1}ab=a^{-1} \right\rangle$ be a nonabelian group of order $8p$, where $p$ is an odd prime number. In this paper, we give the formula to calculate the number of Cayley graphs over $T_{8p}$ up to isomorphism by using the Pólya Enumeration Theorem. Moreover, we get the formula to calculate the number of connected Cayley graphs over $T_{8p}$ by deleting the disconnected graphs. By applying the results, we list the exact number of (connected) Cayley graphs for $3\leq p \leq 13$.
A graph is called an integral graph when all eigenvalues of its adjacency matrix are integers. We study which Cayley graphs over a nonabelian group $$ T_{8n}=\left\langle a,b\mid a^{2n}=b^8=e,a^n=b^4,b^{-1}ab=a^{-1} \right \rangle $$ are integral graphs. Based on the group representation theory, we first give the irreducible matrix representations and characters of $T_{8n}$. Then we give necessary and sufficient conditions for which Cayley graphs over $T_{8n}$ are integral graphs. As applications, we also characterize some families of connected integral Cayley graphs over $T_{8n}$.
It is well established that the entanglement entropy of a critical system generally scales logarithmically with system size. Yet, in this work, we report a new class of non-Hermitian critical transitions that exhibit dramatic divergent dips in their entanglement entropy scaling, strongly violating conventional logarithmic behavior. Dubbed scaling-induced exceptional criticality (SIEC), it transcends existing non-Hermitian mechanisms such as exceptional bound states and non-Hermitian skin effect (NHSE)-induced gap closures, which are nevertheless still governed by logarithmic entanglement scaling. Key to SIEC is its strongly scale-dependent spectrum, where eigenbands exhibit an exceptional crossing only at a particular system size. As such, the critical behavior is dominated by how the generalized Brillouin zone (GBZ) sweeps through the exceptional crossing with increasing system size, and not just by the gap closure per se. We provide a general approach for constructing SIEC systems based on the non-local competition between heterogeneous NHSE pumping directions, and show how a scale-dependent GBZ can be analytically derived to excellent accuracy. Beyond 1D free fermions, SIEC is expected to occur more prevalently in higher-dimensional or even interacting systems, where antagonistic NHSE channels generically proliferate. SIEC-induced entanglement dips generalize straightforwardly to kinks in other entanglement measures such as Renyi entropy, and serve as spectacular demonstrations of how algebraic and geometric singularities in complex band structures manifest in quantum information.
Circularly polarized light (CPL) sources are essential for chiroptics, spintronics, quantum optics, and asymmetric photochemistry. However, conventional approaches fail to simultaneously realize a large luminescence dissymmetry factor (glum) and wide-range tuning of glum in a compact device. Chiral luminophores usually suffer from low glum due to their small molecular sizes. Although chiral metasurfaces can enable a large glum, they lack post-fabrication tunability. Here, we demonstrate that it is possible to achieve high-purity circularly polarized luminescence using achiral metasurfaces. These metasurfaces enable optical tuning and even reversal of luminescence chirality by uncovering and utilizing giant near-field chirality. We validate our concept with upconversion nanoparticles and downshifting dye molecules, experimentally achieving a large glum of up to 1.65, which can be actively and continuously tuned between 1.65 and -1.58. Our approach promises important applications in next-generation CPL sources and detectors, and tunable quantum devices.
Fractional revival, a quantum transport phenomenon critical to entanglement generation in quantum spin networks, generalizes the notion of perfect state transfer on graphs. A Cayley graph $\mathrm{Cay}(G,S)$ is called quasi-abelian if its connection set $S$ is a union of conjugacy classes of the group $G$. In this paper, we establish a necessary and sufficient condition for quasi-abelian Cayley graphs to have fractional revival. This extends a result of Cao and Luo (2022) on the existence of fractional revival in Cayley graphs over abelian groups.
The obtention of quantum-grade rare-earth doped oxide thin films that can be integrated with optical cavities and microwave resonators is of great interest for the development of scalable quantum devices. Among the different growth methods, Chemical Vapour Deposition (CVD) offers high flexibility and has demonstrated the ability to produce oxide films hosting rare-earth ions with narrow linewidths. However, growing epitaxial films directly on silicon is challenging by CVD due to a native amorphous oxide layer formation at the interface. In this manuscript, we investigate the CVD growth of erbium-doped yttrium oxide (Er:Y2O3) thin films on different substrates, including silicon, sapphire, quartz or yttria stabilized zirconia (YSZ). Alternatively, growth was also attempted on an epitaxial Y2O3 template layer on Si (111) prepared by molecular beam epitaxy (MBE) in order to circumvent the issue of the amorphous interlayer. We found that the substrate impacts the film morphology and the crystalline orientations, with different textures observed for the CVD film on the MBE-oxide/Si template (111) and epitaxial growth on YSZ (001). In terms of optical properties, Er3+ ions exhibit visible and IR emission features that are comparable for all samples, indicating a high-quality local crystalline environment regardless of the substrate. Our approach opens interesting prospects to integrate such films into scalable devices for optical quantum technologies.
Let $T_{4p}=\langle a,b\mid a^{2p}=1,a^p=b^2, b^{-1}ab=a^{-1}\rangle$ be the dicyclic group of order $4p$. A Cayley digraph over $T_{4p}$ is called a dicirculant digraph. In this paper, we calculate the number of (connected) dicirculant digraphs of order $4p$ ($p$ prime) up to isomorphism by using the Pólya Enumeration Theorem. Moreover, we get the number of (connected) dicirculant digraphs of order $4p$ ($p$ prime) and out-degree $k$ for every $k$.
In 2018, Chen and Godsil proposed the concept of Laplacian perfect pair state transfer which is a brilliant generalization of Laplacian perfect state transfer. In this paper, we study the existence of Laplacian perfect pair state transfer in the Q-graph of an $r$-regular graph for $r\ge2$. We prove that the Q-graph of an $r$-regular graph does not have Laplacian perfect pair state transfer when $r+1$ is prime or a power of $2$. We also give a sufficient condition for Q-graph to have Laplacian pretty good pair state transfer.
Nuclear spin polarization plays a crucial role in quantum information processing and quantum sensing. In this work, we demonstrate a robust and efficient method for nuclear spin polarization with boron vacancy ($\mathrm{V_B^-}$) defects in hexagonal boron nitride (h-BN) using ground-state level anti-crossing (GSLAC). We show that GSLAC-assisted nuclear polarization can be achieved with significantly lower laser power than excited-state level anti-crossing, making the process experimentally more viable. Furthermore, we have demonstrated direct optical readout of nuclear spins for $\mathrm{V_B^-}$ in h-BN. Our findings suggest that GSLAC is a promising technique for the precise control and manipulation of nuclear spins in $\mathrm{V_B^-}$ defects in h-BN.