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Xiao-Dong Zhang

Publications and source records attributed to Xiao-Dong Zhang.

At least 19 recordsLinked to original sources
arXiv2026

On the Brouwer-type Conjecture for Signless Laplacian Eigenvalues of Graphs

Motivated by Brouwer's conjecture, Ashraf, Omidi and Tayfeh-Rezaie proposed the following Brouwer-type conjecture that for every graph $G$ on $n$ vertices with $m$ edges, the sum $S_k^+(G)$ of its $k$ largest signless Laplacian eigenvalues satisfies $S_k^+(G)\le m+\binom{k+1}{2}$ for $k=1, \ldots, n$. In this paper, we prove that the above conjecture holds. Moreover, the equality holds if and only if $k=1$ and $G$ is either star $K_{1,a}$ or triangle $K_3$ with adding some isolated vertices. For split graphs, properties of block signless Laplacian matrices based on clique and independent set are adapted. While for non-split graphs, some spectral graph substructure are used to control the sum of signless Laplacian eigenvalues.

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arXiv2026

Turán number of a matching and a Berge triangle

For a fixed graph $G$, an $r$-uniform hypergraph is said to contain a Berge-$G$ if there exists a bijection $f\colon E(G)\to E(\mathcal{H})$ for some subhypergraph $\mathcal{H}$ such that $e\subseteq f(e)$ for every $e\in E(G)$. Motivated by Alon and Frankl's study of Turán problems under bounded matching constraints, we investigate the maximum number of edges in $r$-uniform Berge-$K_3$-free hypergraphs with matching number at most~$s$. We determine the exact Turán numbers for the cases $r=3$ and $r=4$. For $r=3$ and $n \geq 3 s$, we prove that every $n$-vertex Berge- $K_3$-free 3-graph with matching number $s$ has at most $s(n-2 s)$ edges, and we characterize the unique extremal hypergraph attaining equality. For $r=4$ and $n \geq 4 s$, the maximum number of edges is $s\lfloor(n-2 s) / 2\rfloor$, except for the exceptional case $s=1$ and $n \equiv 1(\bmod 4)$, in which the bound is $(n-1) / 2$. As a corollary, our results recover the classical theorem of Győri on Berge-$K_3$-free hypergraphs.

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arXiv2026

The saturated spectral radius for complete graphs

A graph is $K_{r+1}$-saturated if it is $K_{r+1}$-free and adding any missing edge creates a copy of $K_{r+1}$. Kim, Kim, Kostochka, and O conjectured that $K_{r-1}\vee(n-r+1)K_1$ minimizes the spectral radius among all $n$-vertex $K_{r+1}$-saturated graphs. They proved the case $r=2$, and the cases $r=3$ and $r\in\{4,5\}$ were subsequently established by Kim, Kostochka, O, Shi, and Wang, and by Wang and Hou, respectively. We settle the conjecture for all $r\ge3$: if $n\ge r+1$ and $G$ is an $n$-vertex $K_{r+1}$-saturated graph, then \[ ρ(G)\ge \frac{r-2+\sqrt{(r-2)^2+4(r-1)(n-r+1)}}{2}, \] with equality if and only if $G\cong K_{r-1}\vee(n-r+1)K_1$. We also prove O's local two-walk conjecture for every $r\ge2$: \[ \sum_{w\in N_G(v)}d_G(w) \ge (r-2)d_G(v)+(r-1)(n-r+1) \qquad(v\in V(G)). \] If $G$ has no universal vertex, the inequality holds with the additional term $(r-1)(r-2)$ on the right-hand side. This constant is best possible uniformly in $n$ for every fixed $r$, and gives a strict improvement when $r\ge3$. A corresponding spectral bound follows.

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arXiv2026

Polynomial positivity cones for Coxeter roots and walks in trees

For a finite simple graph $G$ and an integer $k\ge0$, let $w_k(G)$ denote the number of walks of length $k$. We prove the conjecture of Täubig, Weihmann, Kosub, Hemmecke, and Mayr for every finite tree and determine all equality cases. If $T$ has $n\ge1$ vertices, then $n w_{k+1}(T)-2(n-1)w_k(T)\ge0$ for every $k\ge1$; for $n\ge3$, equality holds if and only if $T$ is a star and $k$ is even, whereas for $n=1$ or $n=2$, equality holds for every $k\ge1$. For non-Dynkin trees and even indices, the proof is based on a polynomial positivity cone associated with the adjacency operator of a finite graph and a positive real root of its simply-laced Coxeter system. For finite connected bipartite non-Dynkin graphs, we establish sufficient positivity conditions in terms of Coxeter orbits and inversion sets, and verify these conditions for indicator roots supported on connected induced subtrees. For non-Dynkin trees, this yields the rooted even-index inequality and, after summation, the corresponding global inequality. We also prove that if $G$ is a finite connected bipartite non-Dynkin simple graph, $\varnothing\ne U\subseteq V(G)$, and the subgraph of $G$ induced by $U$ is a tree, then $|U|w_{k+1}(G,U)-2(|U|-1)w_k(G,U)\ge0$ for every $k\ge0$, where $w_k(G,U)$ counts the length-$k$ walks in $G$ whose initial and terminal vertices lie in $U$; the intermediate vertices are unrestricted. The remaining even-index cases for finite Dynkin trees are handled by generating-function recurrences, while the odd-index cases follow from a spectral covariance identity.

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arXiv2026

The Equality Cases For the Laplacian Conjecture of Brouwer

The Laplacian conjecture of Brouwer asserts that for any graph \(G\) of order n with \(m\) edges, the sum of the \(k\) largest Laplacian eigenvalues satisfies \(s_k(G) \le m + \binom{k+1}{2}\) for $k=1, \ldots, n$. Later, Li and Guo in 2022 further proposed the full Brouwer's Laplacian spectrum conjecture. Recently, Kothari and Tudose in 2026 proved the Brouwer's conjecture. Motivated by their perfect proof and methods, we proved that for a simple graph of order $n$ with $m$ edges and $1\le k\le n-1$, \(s_k(G) = m + \binom{k+1}{2}\) if and only if $G$ is a threshold graph with clique number \(k+1\), which confirms the full Brouwer conjecture proposed by Li and Guo.

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arXiv2026

The Sharp Upper Bounds for the Median Eigenvalues of Graphs

Let $λ_1\geqλ_2\geq\cdots\geqλ_n$ be the eigenvalues of a simple graph $G$ of order $n$. The HL-index of $G$ is defined by $R(G)=\max\|λ_h|,|λ_\ell|\}$ with $h=\lfloor(n+1)/2\rfloor$ and $\ell=\lceil(n+1)/2\rceil$.In this paper, we prove that if $G$ is $ K_4$-minor-free or $ K _ {2,3} $-minor-free, then $R(G)\leq\sqrt{5}-1$ with equality attained by an infinite family of outerplanar graphs.Moreover, we show that $R(G)\leq\sqrt{d-2}$ for triangle-free graphs with maximum degree at most $d$ and average degree at most $(d-2)(d^2-2d+2)/(d^2-3d+5)$.

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arXiv2026

The Equality Cases for the Grone-Merris-Bai Theorem

The Grone--Merris inequality, conjectured by Grone and Merris~(1994) and first proved by Bai~(2011), states that for every graph $G$ of order $n$ and every $1\le k\le n$, $\sum_{i=1}^kλ_i(G)\le\sum_{i=1}^k d_i^*(G)$, where $λ_1\ge\cdots\geλ_n$ are the Laplacian eigenvalues and $d_1^*\ge\cdots\ge d_n^*$ is the conjugate degree sequence. In this paper we determine exactly when equality holds. Using the split-graph trace inequality developed by Kothari and Tudose~(2026) in their proof of Brouwer's Laplacian conjecture---which relies on Bai's theorem and also establishes the equivalence between the two conjectures---together with the recent characterization of the Brouwer equality cases by Cai, Chen, Yang and Zhang~(2027), we prove that equality holds in the Grone--Merris inequality if and only if the graph $G$ belongs to one of two explicitly described families. Both families are obtained from a threshold graph by a surgical operation at one terminal block: in the first family, edges are removed from the initial dominating block; in the second, edges are added inside the initial isolated block. Our analysis yields a complete combinatorial description of all pairs $(G,k)$ for which the Grone--Merris bound is tight.

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arXiv2026

Counting geodesic paths in graphs

A geodesic is a shortest path which connects a pair of vertices of a graph G. In this paper we define the geodesic subpath number gpn(G) of a graph G as the number of geodesics in G. The number of subtrees and subpaths are already studied in literature, but they are both large quantities. Hence, the geodesic subpath number which is related to these quantities but smaller than both, seems worthy of investigation. We first consider extremal graphs with respect to the geodesic subpath number among all connected graphs on n vertices. This number is minimized by the so called geodetic graphs, i.e. graphs in which each pair of vertices is connected by precisely one geodesic. As for the graphs which maximize the geodesic subpath number, we provide an upper bound on gpn(G) in terms of n and we further consider several graph families which might have a large gpn(G). Yet, their value of gpn(G) still does not attain the established bound, so narrowing the gap remains as an open problem. We also consider the class of cactus graphs on n vertices and k cycles and among them characterize extremal graphs with respect to this new invariant.

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arXiv2026

Sharp Bounds for Guiduli-Type Hereditary Spectral Problems

Guiduli asked in 1996 the following problem concerning the maximum spectral radius of a graph under hereditary density constraints. If an $n$-vertex graph $G$ satisfies $e(H)\le c|V(H)|^2$ for every subgraph $H$ of $G$, must one have $λ(G)\le 2cn$? More generally, what remains true when the exponent $2$ is replaced by a constant less than $2$? We study the natural power-law version of this question for all $1<p\le2$. For $1<p\le 2$, define \[ d_p(G)=\max_{\varnothing\ne S\subseteq V(G)}\frac{e(G[S])}{|S|^p}. \] We determine the sharp asymptotic upper bound for $λ(G)$ in terms of $d_p(G)$ and $n$. More precisely, every $n$-vertex graph $G$ with at least one edge satisfies \[ λ(G)\le \begin{cases} \left(\left(\max_{t\in\mathbb N_{\ge1}}\dfrac{t}{(t+1)^p}\right)^{-1}+o(1)\right)d_p(G)\sqrt n,&1<p<3/2,\\[0.4em] \left(\dfrac{3\sqrt3}{4}+o(1)\right)d_p(G)\sqrt{n\log n},&p=3/2,\\[0.4em] (\mathfrak C_p+o(1))d_p(G)n^{p-1},&3/2<p<2, \end{cases} \] and each constant here is best possible. Here $\mathfrak C_p$ is characterized by an exact variational problem over finite kernels. We apply a sparse graphon operator estimate to convert hereditary $p$-density bounds into sharp spectral bounds, and this estimate also explains the transition at the critical exponent $p=3/2$. For the endpoint $p=2$, Wilf's theorem gives the exact finite-$n$ bound $λ(G)\le 2d_2(G)n$, with equality for $K_n$. Thus Guiduli's power-law problem is resolved in its sharp asymptotic form for every $1<p\leq2$, including exact leading constants.

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arXiv2026

Quantum Implicit-Explicit Schemes for Multiscale Ordinary and Partial Differential Equations via Schrödingerization

In this paper, we present a quantum implicit-explicit (IMEX) scheme for multiscale ordinary and partial differential equations whose discretization parameters are independent of the scaling parameter $\varepsilon$. A key ingredient of our approach is a continuous-time formulation of classical IMEX schemes, which decouples the evolution time of the quantum algorithm from the physical time of the differential equation and is therefore particularly useful in multiscale settings. Building on this idea, we employ the Schrödingerization framework [Phys. Rev. Lett. 133 (2024), 230602] to implement IMEX schemes on quantum computers. Compared to previous HHL type quantum AP scheme [J. Comput. Phys. 471 (2022), 111641], this new method requires narrower -- an extra logarithmic factor -- auxiliary register numerical examples on linear heat and multiscale telegraph equations demonstrate the independence in $\varepsilon$ of the method.

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arXiv2026

Polylogarithmic Bounds for Nested Cycles without Geometric Crossings

A problem of Erdős asks for extremal conditions forcing edge-disjoint cycles with a prescribed nested structure. In the geometric version, the nesting is required to be noncrossing with respect to the cyclic orders. Fernández, Kim, Kim and Liu proved that constant average degree forces two such cycles. We prove a polylogarithmic bound for the natural multi-layer version: for every fixed $k\ge 3$, every sufficiently large $n$-vertex graph with at least \[ C_k n(\log n)^{k-1}(\log\log n)^{k-3} \] edges contains $k$ pairwise edge-disjoint nested cycles without geometric crossings. The proof combines the robust sublinear expander framework of Alon, Bucić, Sauermann, Zakharov and Zamir with a controlled wrapping lemma that permits the layers to be built successively with controlled length.

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arXiv2026

Positive and negative 3-energies of graphs

For a simple graph $G$ with $n$ vertices, let $A_G$ denote the adjacency matrix of $G$, and let $λ_1(G) \geq λ_2(G) \geq \dots \geq λ_n(G)$ be its eigenvalues. For an integer $p \geq 2$, the positive $p$-energy and negative $p$-energy of $G$, denoted $\mathcal{E}^+_p(G)$ and $\mathcal{E}^-_p(G)$, are defined as follows: $\mathcal{E}^+_p(G) = \sum_{λ_i(G) > 0} |λ_i(G)|^p$ and $\mathcal{E}^-_p(G) = \sum_{λ_i(G) < 0} |λ_i(G)|^p,$ respectively. Tang, Liu, and Wang proposed a conjecture that, for any integer $p \geq 2$, every connected $n$-vertex graph $G$ satisfies $\mathcal{E}^+_p(G) \geq \mathcal{E}^+_p(P_n)$. Akbari, Kumar, Mohar, and Pragada conjectured that, for any $p \geq 2$, every connected $n$-vertex graph $G$ satisfies $\mathcal{E}^-_p(G) \geq \mathcal{E}^-_p(K_n)$, and they proved this conjecture for $p \geq 4$. In this paper, we prove that every connected $n$-vertex graph, except for $K_1$, $K_2$, and $P_3$, satisfies $\mathcal{E}^+_3(G) \geq \frac{\sqrt{5}}{2}n$. Moreover, we show that for any integer $p \geq 3$, every connected $n$-vertex graph $G$ satisfies $\mathcal{E}^-_p(G) \geq \mathcal{E}^-_p(K_n)$, which improves upon the previously known result.

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arXiv2026

Turán extremal graphs vs. Signless Laplacian spectral Turán extremal graphs

Let $F$ be a graph with chromatic number $χ(F) = r+1$. Denote by $ex(n, F)$ and $Ex(n, F)$ the Turán number and the set of all extremal graphs for $F$, respectively. In addition, $ex_{ssp}(n, F)$ and $Ex_{ssp}(n, F)$ are the maximum signless Laplacian spectral radius of all $n$-vertex $F$-free graphs and the set of all $n$-vertex $F$-free graphs with signless Laplacian spectral radius $ex_{ssp}(n, F)$, respectively. It is known that $Ex_{ssp}(n, F)\supset Ex(n, F)$ if $F$ is a triangle. In this paper, employing the regularity method and Füredi's stability theorem, we prove that for a given graph $F$ and $r\geqslant 3$, if $ex(n, F) = t_r(n)+O(1)$, then $ Ex_{ssp}(n, F) \subseteq Ex(n, F)$ for sufficiently large $n$, where $t_r(n)$ is the number of edges in the Turán graph $T_r(n)$.

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arXiv2025

A hypergraph analogue of Alon-Frankl Theorem

Recently, Alon and Frankl (JCTB, 2024) determined the maximum number of edges in $K_{\ell+1}$-free $n$-vertex graphs with bounded matching number. For integers $\ell\ge r \ge 2$, the family $\mathcal{K}_{\ell+1}^{r}$ consists of all $r$-graphs $F$ with at most $\binom{\ell+1}{2}$ edges such that, for some $(\ell+1)$-set $K$, every pair $\{x,y\} \subseteq K$ is covered by an edge in $F$. In this paper, we study the maximum number of edges in $\mathcal{K}_{\ell+1}^r$-free $r$-uniform hypergraphs that have the matching number at most $s$, that is, $\mathrm{ex}_r(n, \{\mathcal{K}_{\ell+1}^r, M^r_{s+1}\})$, and obtain the exact value for sufficiently large $n$, along with the corresponding extremal hypergraph. This result can be viewed as a hypergraph extension of the work of Alon and Frankl. In addition, for the $3$-uniform Fano plane $\mathbb{F}$, we determine the exact value of $\mathrm{ex}_3(n, \{\mathbb{F}, M^3_{s+1}\})$, and characterize the corresponding extremal hypergraph.

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arXiv2025

Verifcation of general multi-qudit pure states

Verifying prepared quantum states is crucial for hybrid systems whose subsystems may have different local dimensions. We present a generalized stabilizer framework and associated test that apply to general multi-qudit states, including composite-dimensional and hybrid architectures. Using only adaptive local measurements, our method verifies qutrit-qubit states, arbitrary two-qubit pure states, Bell/Bell-like, GHZ/GHZ-like, graph, hypergraph, multigraph, and multihypergraph states, with efficiencies matching or surpassing the best known schemes.

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arXiv2025

Extremal graphs for the maximum $A_α$-spectral radius of graphs with order and size

In 1986, Brualdi and Solheid firstly proposed the problem of determining the maximum spectral radius of graphs in the set $\mathcal{H}_{n,m}$ consisting of all simple connected graphs with $n$ vertices and $m$ edges, which is a very tough problem and far from resolved. The $A_α$-spectral radius of a simple graph of order $n$, denoted by $ρ_α(G)$, is the largest eigenvalue of the matrix $A_α(G)$ which is defined as $αD(G)+(1-α)A(G)$ for $0\le α< 1$, where $D(G)$ and $A(G)$ are the degree diagonal and adjacency matrices of $G$, respectively. In this paper, if $r$ is a positive integer, $n>30r$ and $n-1\leq m \le rn-\frac{r(r+1)}{2}$, we characterize all extremal graphs which have the maximum $A_α$-spectral radius of graphs in the set $\mathcal{H}_{n,m}$. Moreover, the problem on $A_α$-spectral radius proposed by Chang and Tam [T.-C. Chang and B.-T. Tam, Graphs of fixed order and size with maximal $A_α$-index. Linear Algebra Appl. 673 (2023), 69-100] has been solved.

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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

Quantum Algorithms for Solving Generalized Linear Systems via Momentum Accelerated Gradient and Schrödingerization

In this paper, we propose a quantum algorithm that combines the momentum accelerated gradient method with Schrödingerization [S. Jin, N. Liu and Y. Yu, Phys. Rev. Lett, 133 (2024), 230602][S. Jin, N. Liu and Y. Yu, Phys. Rev. A, 108 (2023), 032603], achieving polynomial speedup over its classical counterpark in solving linear systems. The algorithm achieves a query complexity of the same order as the Schrödingerization based damped dynamical system method, namely, linear dependence on the condition number of the matrix, and can overcome the practical limitations of existing non-Schrödingerization-based quantum linear system algorithms. These limitations stem from their reliance on techniques such as VTAA and RM, which introduce substantial quantum hardware resource overhead. Furthermore, it demonstrates both theoretically and experimentally that the auxiliary variables introduced by our method do not dominate the error reduction at any point, thereby preventing a significant increase in the actual evolution time compared to the theoretical prediction. In contrast, the damped method fails to meet this criterion. This gives new perspectives for quantum algorithms for linear systems, establishing a novel analytical framework for algorithms with broader applicability, faster convergence rates, and superior solution quality.

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