arXiv subjects

Yi-Zheng Fan

Publications and source records attributed to Yi-Zheng Fan.

At least 19 recordsLinked to original sources

arXiv2026

Maximizing the signless Laplacian spectral radius of simplicial 2-complexes with a prescribed second Betti number

We study how a prescribed second Betti number constrains the largest eigenvalue of the signless Laplacian on edges of a pure two-dimensional simplicial complex. For each fixed positive second Betti number, we determine all maximizing complexes provided that the number of vertices is sufficiently large. Every maximizer consists of a full cone over a complete graph together with a family of triangles avoiding the apex, any two of which share an edge; the number of added triangles equals the prescribed Betti number. The maximizer is unique up to isomorphism except when this number is three or four, for which we describe all additional extremal complexes. The proof uses homological constraints and Perron vector estimates to establish the full cone structure, followed by an exact Schur complement comparison to classify the added triangles. We also obtain an asymptotic expansion of the maximum spectral radius and extend the extremal result to Betti numbers growing more slowly than the fourth root of the number of vertices. These results provide a two-dimensional counterpart of spectral extremal theorems for connected graphs with a prescribed cyclomatic number.

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arXiv2026

The multiplicity of eigenvalues of nonnegative tensors and hypergraphs

Hu and Ye conjectured that for an $n$-dimensional tensor $\mathcal{A}$ of order $k$ with an eigenvalue $λ$ and the corresponding eigenvariety $\mathcal{V}_λ(\mathcal{A})$, the algebraic multiplicity $\mathrm{am}(λ)$ of $λ$ satisfies: $$\mathrm{am}(λ) \ge \sum_{i=1}^κ\dim(V_i)(k-1)^{\dim(V_i)-1},$$ where $V_1,\ldots,V_κ$ are all irreducible components of $\mathcal{V}_λ(\mathcal{A})$. In this paper, we establish that for any weakly irreducible nonnegative tensor $\mathcal{A}$ with spectral radius $ρ$, all the eigenvalues $λ$ of $\mathcal{A}$ with $|λ|=ρ$ satisfy $\mathrm{am}(λ) \ge |\mathbb{V}_λ(\mathcal{A})|$, where $\mathbb{V}_λ(\mathcal{A})$ is the projective eigenvariety of $\mathcal{A}$ associated with $λ$. As a direct consequence, we confirm the Hu-Ye Conjecture for two classes of eigenvalues: (1) all eigenvalues of weakly irreducible nonnegative tensors with modulus equal to the spectral radius, and (2) the least H-eigenvalues of weakly irreducible $Z$-tensors. Furthermore, we characterize the equality condition in Hu-Ye's conjecture for the eigenvalues of several hypergraph classes and present a new conjecture regarding the eigenvalue multiplicities of hypergraphs. As an initial step toward this new conjecture, we give a sufficient condition for a point in the projective eigenvariety of a tensor to have local intersection multiplicity one.

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arXiv2026

Tensor Spectral Stability for Uniform Hypergraphs with Bounded Matching Number

We establish a tensor spectral stability theorem for uniform hypergraphs with bounded matching number. More precisely, for fixed integers $k\geq 3$ and $β\geq2$, and sufficiently large $n$, we prove that every $n$-vertex $k$-uniform hypergraph $H$ with matching number at most $β$ and tensor spectral radius close to the maximum possible value among all such hypergraphs must be structurally close to the extremal hypergraph $S_{n,k,β}$, whose edges consist of all $k$-sets intersecting a fixed set of $β$ vertices. Furthermore, we show that every edge of $H$ intersects this distinguished vertex set and that $H$ contains all but a small proportion of the edges of $S_{n,k,β}$. As an application, we obtain a new proof of the spectral version of the Erdős matching conjecture for sufficiently large $n$.

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arXiv2026

Degree Majorization and Laplacian Eigenvalue Sums for Simplicial Complexes

Let $K$ be an $r$-dimensional simplicial complex. We prove that the spectrum of its $(r - 1)$-dimensional up-Laplacian is majorized by the conjugate degree sequence of its $(r - 1)$-dimensional faces: \[ {\mathbfλ}_{r-1}(K) \preccurlyeq {\mathbf d}_{r-1}^\top(K). \] We also establish a Brouwer-type inequality: for every integer $\ell \geq 1$, \[ \sum_{i = 1}^{\ell}λ_{r-1,i}(K) \leq \frac{r + 1}{2}f_r(K) + \frac{f_{r - 2}(K)}{r} \binom{\ell + 1}{2}, \] where $λ_{r-1,i}(K)$ denotes the $i$-th largest eigenvalue in the spectrum ${\mathbfλ}_{r-1}(K)$, and $f_t(K)$ denotes the number of $t$-dimensional faces of $K$. These results provide higher-dimensional analogs of the Grone-Merris-Bai theorem and the Brouwer-Kothari-Tudose theorem and recover the corresponding graph results when $r=1$. We show that the Duval-Reiner conjecture on the majorization by the conjugate degree sequence of vertices fails in every dimension $r \geq 2$. More precisely, for every $n \geq r + 5$, we construct a pure $r$-dimensional complex on $n$ vertices that violates the conjectured inequality at the fifth partial sum.

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arXiv2026

Degree-similar Unicyclic Graphs are Isomorphic

Two graphs are degree-similar if their adjacency matrices and degree matrices are simultaneously similar. Godsil and Sun asked whether non-isomorphic degree-similar unicyclic graphs exist. We prove that they do not exist. Every graph degree-similar to a unicyclic graph is isomorphic to it. The proof uses a symbolic leaf-peeling algorithm to recover the rooted trees attached to the unique cycle, and a rigidity lemma for colored cycles to identify automorphisms of the cycle.

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arXiv2026

Signless Laplacian Spectral Radius and Link Homology of Simplicial Complexes

In this paper, we study the signless Laplacian spectral radius of pure simplicial complexes under local homological restrictions on links. Let $K$ be a pure $r$-dimensional complex on $n$ vertices, ${\mathfrak q}_{r-1}(K)$ be the spectral radius of the $(r-1)$-up signless Laplacian of $K$, and ${\operatorname{lk}}_K(σ)$ be the link of a face $σ$ in $K$. We prove that if the homology $\widetilde H_t({\operatorname{lk}}_K(σ), {\mathbb R})=0$ for every face $σ\in K$ with $|σ|=r-t$, then \[ {\mathfrak q}_{r-1}(K)\le tn-(t-1)(r+1).\] Moreover, if $K$ is $r$-down path connected and $n\ge r+2+\binom{r+1}{t}\binom{r}{t}$, equality holds if and only if $K \cong Δ_{r+1-t} \star Δ_{n-r-1+t}^{t}$, where $Δ_n$ denotes a simplex on $n$ vertices, $Δ_n^{p}$ denotes the $(p-1)$-skeleton of $Δ_n$, and $\star$ denotes the join of two complexes.

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arXiv2026

Upper Bounds for the Largest Laplacian Eigenvalue of Simplicial Complexes

Let $K$ be a finite $r$-dimensional simplicial complex with vertex set $V$ of size $n$. We study the largest eigenvalue of the combinatorial $(r-1)$-up Laplacian $L^{\operatorname{up}}_{r-1}(K)$. It is known that \[ λ_{\max}\bigl(L^{\operatorname{up}}_{r-1}(K)\bigr)\le n. \] We first give a homological equality criterion for this universal bound, namely, the equality holds if and only if the $r$-dimensional complement $K^c$ of $K$ has a nonzero reduced homology $\widetilde H_{r-1}(K^c,\mathbb{R})$. For $r=1$, this is the classical graph condition that the complement graph is disconnected. Secondly, we prove a sharper upper bound for $λ_{\max}(L^{\operatorname{up}}_{r-1}(K))$: \[ λ_{\max}(L^{\operatorname{up}}_{r-1}(K)) \le \max_{F\in S_r(K)} \bigl|\bigcup_{E \in \partial F} N_K(E) \bigr| \le n,\] where, for an $(r-1)$-face $E$, $N_K(E)$ denotes the set of vertices $u$ outside $E$ such that the union $E \cup \{u\}$ is an $r$-face of $K$. This is the high-dimensional analog of the graph Laplacian bound. We give an explicit characterization of the equality case, and construct a broad family attaining the bound, namely, the partite semiregular complexes with admissible additions.

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arXiv2026

Orthogonal degree-similarity of edge-deleted strongly regular graphs

Godsil and Sun asked whether, for a strongly regular graph $X$ and any two different edges $e$ and $f$, the edge-deleted graphs $X\setminus e$ and $X\setminus f$ are degree-similar. We give an affirmative answer to the problem of Godsil and Sun. In fact, we prove the stronger statement that if $X$ is a $1$-walk-regular graph, then for any two edges $e$ and $f$ of $X$, the graphs $X\setminus e$ and $X\setminus f$ are orthogonally degree-similar. The proof is based on an edge version of the orthogonal-intertwiner method: the equality of the Gram matrices of the projected endpoint vectors in every eigenspace yields an orthogonal matrix commuting with the adjacency matrix and sending one pair of ordered endpoint vectors to the other.

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arXiv2026

Smith normal forms for coalescences at cospectral vertices

Let $L_μ(G)=A(G)-μD(G)$ be the generalized $μ$-adjacency matrix of a finite graph $G$. Fan, Xing, Zhang, and Wang constructed pairs of non-degree-similar trees for which the Smith normal forms of the matrices $tI-L_μ(G)$ over $\mathbb{Q}(μ)[t]$ coincide, and conjectured that their construction remains valid when the attached rooted path is replaced by an arbitrary rooted tree. We prove this conjecture as a consequence of a more general coalescence theorem: if a finite graph $H$ has two vertices $u$ and $v$ that are cospectral for $L_μ(H)$, then, for every finite rooted graph $R$ with root $r$, the matrices \[ tI-L_μ(R(r)\odot H(u)) \quad\text{and}\quad tI-L_μ(R(r)\odot H(v)) \] have the same Smith normal form over $\mathbb{Q}(μ)[t]$, where $\odot$ denotes coalescence of rooted graphs. The proof uses an orthogonal intertwiner over a real closed extension field.

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arXiv2026

Signless Laplacian spectral radius of simplicial complexes without holes

We study a spectral analog of the Turán problem for simplicial complexes. Specifically, we consider the extremal problem of maximizing the signless Laplacian spectral radius among simplicial complexes without holes. We determine the structure of the simplicial complex attaining the maximum spectral radius, extending classical extremal results for graphs without cycles to the setting of higher-dimensional simplicial complexes. More generally, we establish an upper bound on the signless Laplacian spectral radius of simplicial complexes with prescribed Betti numbers. As an application, using the connection between the signless Laplacian spectral radius and the face numbers of a simplicial complex, we derive bounds on Turán numbers for both hypergraphs and simplicial complexes. Our technique involves the canonical Alexander dual of perfect matchings and coloring of simplicial complexes.

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arXiv2026

Generalized spectral Turán problems for disjoint cliques

The generalized Turán number $\text{ex}(n, H, F)$ denotes the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Let $kK_{r+1}$ be the disjoint union of $k$ copies of the complete graph $K_{r+1}$. Recently, Gerbner determined $\text{ex}(n, K_{t},kK_{r+1})$ for all sufficiently large $n$. In this paper, we study a spectral analogue of this problem via the $t$-clique tensor of a graph. We prove that if an $n$-vertex $kK_{r+1}$-free graph $G$ maximizes the $t$-clique spectral radius, then for sufficiently large $n$, $G$ is the join of a complete graph $K_{k-1}$ and the $r$-partite Turán graph $T_{r}(n-k+1)$. This establishes a spectral counterpart of Gerbner's Theorem. Moreover, in the case $t=2$, our result recovers a theorem of Ni, Wang, and Kang on the maximum spectral radius of $kK_{r+1}$-free graphs.

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arXiv2026

Signless Laplacian spectral radius of simplicial complexes without $r$-dimensional wheels

An $r$-dimensional wheel is defined as the join of an $(r-2)$-simplex and a cycle. In this paper, we study the maximum signless Laplacian spectral radius of $n$-vertex $r$-dimensional pure simplicial complexes that contain no $r$-dimensional wheels. For sufficiently large $n$, we determine the extremal complexes that attain this maximum. Our result generalizes the corresponding extremal results of signless Laplacian on graphs and provides a spectral anlogue of a theorem of Sós, Erdős and Brown on the maximum number of facets of simplicial complexes in the case $r=2$.

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arXiv2026

Some Turán-type results for the signless Laplacian spectral radius

Half a century ago, Bollobás and Erdős [Bull. London Math. Soc. 5 (1973)] proved that every $n$-vertex graph $G$ with $e(G)\ge (1- \frac{1}{k} + \varepsilon )\frac{n^2}{2}$ edges contains a blowup $K_{k+1}[t]$ with $t=Ω_{k,\varepsilon}(\log n)$. A well-known theorem of Nikiforov [Combin. Probab. Comput. 18 (3) (2009)] asserts that if $G$ is an $n$-vertex graph with adjacency spectral radius $λ(G)\ge (1- \frac{1}{k} + \varepsilon)n$, then $G$ contains a blowup $K_{k+1}[t]$ with $t=Ω_{k,\varepsilon}(\log n)$. This gives a spectral version of the Bollobás--Erdős theorem. In this paper, we systematically explore variants of Nikiforov's result in terms of the signless Laplacian spectral radius, extending the supersaturation, blowup of cliques and the stability results.

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arXiv2025

Spectral Turán problems for nondegenerate hypergraphs

Keevash, Lenz and Mubayi developed a general criterion for hypergraph spectral extremal problems in their seminal work (SIAM J. Discrete Math., 2014). Their framework shows that extremal results on the $α$-spectral radius (for $α> 1$) may be deduced from a corresponding hypergraph Turán problem exhibiting stability properties, provided its extremal construction satisfies certain continuity assumptions. In this paper, we establish a spectral stability result for nondegenerate hypergraphs, extending the Keevash--Lenz--Mubayi criterion. Applying this result, we derive two general spectral Turán theorems for hypergraphs with bipartite or multipartite pattern, thereby transforming spectral Turán problems into the corresponding purely combinatorial problems related to degree-stability in nondegenerate $k$-graph families. As applications, we determine the maximum $α$-spectral radius for several classes of hypergraph and characterize the corresponding extremal hypergraphs, such as the expansion of complete graphs, the generalized fans, the cancellative hypergraphs, the generalized triangles, and a special book hypergraph.

math.CO↗

arXiv2025

On the Spread of Graph-Related Matrices

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 $A_α$-matrix of graphs. The $A_α$-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 $A_α (G) = αD(G) + (1-α) A(G)$. Let $λ_1^{(α)} (G)$ and $λ_n^{(α)} (G)$ denote the largest and smallest eigenvalues of $A_α (G)$, respectively. We determined the unique graph that maximizes $λ^{(α)}_1 (G) - β\cdotλ^{(γ)}_n (G)$ among all connected $n$-vertex graphs for sufficiently large $n$, where $0 \leq α< 1$, $1/2\leq γ< 1$ and $0<βγ\leq 1$. 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 $α= γ= 1/2$ and $β= 1$.

math.CO↗

arXiv2025

Degree-similar graphs and cospectral graphs

Let $G$ be a graph with adjacency matrix $A(G)$ and degree matrix $D(G)$, and let $L_μ(G):=A(G)-μD(G)$. Two graphs $G_1$ and $G_2$ are called \emph{degree-similar} if there exists an invertible matrix $M$ such that $M^{-1} A(G_1) M =A(G_2)$ and $M^{-1} D(G_1) M =D(G_2)$. In this paper, we address three problems concerning degree-similar graphs proposed by Godsil and Sun. First, we present a new characterization of degree-similar graphs using degree partition, from which we derive methods and examples for constructing cospectral graphs and degree-similar graphs. Second, we construct infinite pairs of non-degree-similar trees $G_1$ and $G_2$ such that $tI- L_μ(G_1)$ and $tI-L_μ(G_2)$ have the same Smith normal form over $\Q(μ)[t]$, which provides a negative answer to a problem posed by Godsil and Sun. Third, we establish several invariants of degree-similar graphs and obtain results on unicyclic graphs that are degree-similar determined. Lastly we prove that for a strongly regular graph $G$ and any two edges $e$ and $f$ of $G$, $G \backslash e$ and $G \backslash f$ have identical $μ$-polynomial, i.e., $\det(tI-L_μ(G \backslash e))=\det(tI-L_μ(G \backslash f))$, which enables the construction of pairs of non-isomorphic graphs with same $μ$-polynomial, where $G \backslash e$ denotes the graph obtained from $G$ by deleting the edge $e$.

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arXiv2025

The spectra of Laplace operators on covering simplicial complexes

We give a decomposition of the Laplace operator (in matrix form) of a covering simplicial complex as a direct sum of several matrices, one of which is the Laplace operator of the base complex. It follows that the spectrum of a covering simplicial complex is a multiset union of the spectrum of the base simplicial complex and the spectra of other relevant matrices, which implies the spectral inclusion property of Horak and Jost. In the case of a $2$-fold covering, we show that the spectrum is a multiset union of the spectrum of the base complex and that of an incidence-signed simplicial complex, thereby generalizing a result of Bilu and Linial from graphs to simplicial complexes. Additionally, we show that the dimension of the cohomology of a covering complex is greater than or equal to that of the base complex. Our arguments exploit the coverings of incidence graphs of simplicial complexes and the representation theory of permutation groups.

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arXiv2025

Spectral bipartite Turan problems on linear hypergraphs

Let $F$ be a graph, and let $\mathcal{B}_r(F)$ be the class of $r$-uniform Berge-$F$ hypergraphs. In this paper, we establish a relationship between the spectral radius of the adjacency tensor of a uniform hypergraph and its local structure through walks. Based on the relationship, we give a spectral asymptotic bound for $\mathcal{B}_{r}(C_3)$-free linear $r$-uniform hypergraphs and upper bounds for the spectral radii of $\mathcal{B}_{r}(K_{2,t})$-free or $\{\mathcal{B}_{r}(K_{s,t}),\mathcal{B}_{r}(C_{3})\}$-free linear $r$-uniform hypergraphs, where $C_{3}$ and $K_{s,t}$ are respectively the triangle and the complete bipartite graph with one part having $s$ vertices and the other part having $t$ vertices. Our work implies an upper bound for the number of edges of $\{\mathcal{B}_{r}(K_{s,t}),\mathcal{B}_{r}(C_{3})\}$-free linear $r$-uniform hypergraphs and extends some of the existing research on (spectral) extremal problems of hypergraphs.

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