arXiv subjects

Yi-Zheng Fan

Publications and source records attributed to Yi-Zheng Fan.

At least 37 records · Page 2Linked to original sources

arXiv2024

The largest normalized Laplacian eigenvalue and incidence balancedness of simplicial complexes

Let $K$ be a simplicial complex, and let $Δ_i^{up}(K)$ be the $i$-th up normalized Laplacian of $K$. Horak and Jost showed that the largest eigenvalue of $Δ_i^{up}(K)$ is at most $i+2$, and characterized the equality case by the orientable or non-orientable circuits. In this paper, by using the balancedness of signed graphs, we show that $Δ_i^{up}(K)$ has an eigenvalue $i+2$ if and only if $K$ has an $(i+1)$-path connected component $K'$ such that the $i$-th signed incidence graph $B_i(K')$ is balanced, which implies Horak and Jost's characterization. We also characterize the multiplicity of $i+2$ as an eigenvalue of $Δ_i^{up}(K)$, which generalizes the corresponding result in graph case. Finally we gave some classes of infinitely many simplicial complexes $K$ with $Δ_i^{up}(K)$ having an eigenvalue $i+2$ by using wedge, Cartesian product and duplication of motifs.

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arXiv2024

Spectral Methods for Matrix Product Factorization

A graph $G$ is factored into graphs $H$ and $K$ via a matrix product if there exist adjacency matrices $A$, $B$, and $C$ of $G$, $H$, and $K$, respectively, such that $A = BC$. In this paper, we study the spectral aspects of the matrix product of graphs, including regularity, bipartiteness, and connectivity. We show that if a graph $G$ is factored into a connected graph $H$ and a graph $K$ with no isolated vertices, then certain properties hold. If $H$ is non-bipartite, then $G$ is connected. If $H$ is bipartite and $G$ is not connected, then $K$ is a regular bipartite graph, and consequently, $n$ is even. Furthermore, we show that trees are not factorizable, which answers a question posed by Maghsoudi et al.

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arXiv2024

The largest Laplacian eigenvalue and the balancedness of simplicial complexes

Let $K$ be a simplical complex, and let $\mathcal{L}_i^{up}(K), \mathcal{Q}_i^{up}(K)$ be the $i$-th up Laplacian and signless Laplacian of $K$, respectively. In this paper we proved that the largest eigenvalue of $\mathcal{L}_i^{up}(K)$ is not greater than the largest eigenvalue of $\mathcal{Q}_i^{up}(K)$; furthermore, if $K$ is $(i+1)$-path connected, then the equality holds if and only if the $i$-th incidence signed graph $B_i(K)$ of $K$ is balanced. As an application we provided an upper bound for the largest eigenvalue of the $i$-th up Laplacian of $K$, which improves the bound given by Horak and Jost and generalizes the result of Anderson and Morley on graphs.We characterized the balancedness of simplicial complexes under operations such as wedge sum, join, Cartesian product and duplication of motifs. For each $i \ge 0$, by using wedge sum or duplication of motifs, we can construct an infinitely many $(i+1)$-path connected simplicial complexes $K$ with $B_i(K)$ being balanced.

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arXiv2024

Hypergraph coverings and Ramanujan Hypergraphs

In this paper we investigate Ramanujan hypergraphs by using hypergraph coverings. We first show that the spectrum of a $k$-fold covering $\bar{H}$ of a connected hypergraph $H$ contains the spectrum of $H$, and that it is the union of the spectrum of $H$ and the spectrum of an incidence-signed hypergraph with $H$ as underlying hypergraph if $k=2$, which generalizes Bilu-Linial result on graph coverings. We give a lower bound for the second largest eigenvalue of a $d$-regular hypergraph by universal cover, which generalizes Alon-Boppana bound on $d$-regular graphs and Feng-Li bound on $(d,r)$-regular hypergraphs. By using interlacing family of polynomials, we prove that every $(d,r)$-regular hypergraph has a right-sided Ramanujan $2$-covering, and has a left-sided Ramanujan $2$-covering if the roots of the matching polynomial of its incident graph satisfy some condition. By Ramanujan $2$-coverings, we prove the existence of some families of infinite many left-sided or right-sided $(d,r)$-regular Ramanujan hypergraphs under certain conditions on $d$ and $r$.

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arXiv2023

The spectral property of hypergraph coverings

Let $H$ be a connected $m$-uniform hypergraph, and let $\mathcal{A}(H)$ be the adjacency tensor of $H$ whose spectrum is simply called the spectrum of $H$. Let $s(H)$ denote the number of eigenvectors of $\mathcal{A}(H)$ associated with the spectral radius, and $c(H)$ denote the number of eigenvalues of $\mathcal{A}(H)$ with modulus equal to the spectral radius, which are respectively called the stabilizing index and cyclic index of $H$. Let $\bar{H}$ be a $k$-fold covering of $H$ which can be obtained from some permutation assignment in the symmetric group $\mathbf{S}_k$ on $H$. In this paper, we first characterize the connectedness of $\bar{H}$ by its incidence graph and the permutation assignment, and then investigate the relationship between the spectral property of $H$ and that of $\bar{H}$. By applying module theory and group representation, if $\bar{H}$ is connected, we prove that $s(H) \mid s(\bar{H})$ and $c(H) \mid c(\bar{H})$. In particular, when $\bar{H}$ is a $2$-fold covering of $H$, if $m$ is even, we show that regardless of multiplicities, the spectrum of $\bar{H}$ contains the spectrum of $H$ and the spectrum of a signed hypergraph with $H$ as underlying hypergraph; if $m$ is odd, we give an explicit formula for $s(\bar{H})$. We also find some differences on the spectral property between hypergraph coverings and graph coverings by examples.

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arXiv2023

Linear spectral Turan problems for expansions of graphs with given chromatic number

An $r$-uniform hypergraph is linear if every two edges intersect in at most one vertex. The $r$-expansion $F^{r}$ of a graph $F$ is the $r$-uniform hypergraph obtained from $F$ by enlarging each edge of $F$ with a vertex subset of size $r-2$ disjoint from the vertex set of $F$ such that distinct edges are enlarged by disjoint subsets. Let $ex_{r}^{lin}(n,F^{r})$ and $spex_{r}^{lin}(n,F^{r})$ be the maximum number of edges and the maximum spectral radius of all $F^{r}$-free linear $r$-uniform hypergraphs with $n$ vertices, respectively. In this paper, we present the sharp (or asymptotic) bounds of $ex_{r}^{lin}( n,F^{r})$ and $spex_{r}^{lin}(n,F^{r})$ by establishing the connection between the spectral radii of linear hypergraphs and those of their shadow graphs, where $F$ is a $(k+1)$-color critical graph or a graph with chromatic number $k$.

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arXiv2022

The trace of uniform hypergraphs with application to Estrada index

In this paper we investigate the traces of the adjacency tensor of hypergraphs (simply called the traces of hypergraphs). We give new expressions for the traces of hypertrees and linear unicyclic hypergraphs by the weight function assigned to their connected sub-hypergraphs, and provide some perturbation results for the traces of a hypergraph with cut vertices. As applications we determine the unique hypertree with maximum Estrada index among all hypertrees with fixed number of edges and perfect matchings, and the unique unicyclic hypergraph with maximum Estrada index among all unicyclic hypergraph with fixed number of edges and girth $3$.

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arXiv2022

High-ordered spectral characterization of unicyclic graphs

In this paper we will apply the tensor and its traces to investigate the spectral characterization of unicyclic graphs. Let $G$ be a graph and $G^m$ be the $m$-th power (hypergraph) of $G$. The spectrum of $G$ is referring to its adjacency matrix, and the spectrum of $G^m$ is referring to its adjacency tensor. The graph $G$ is called determined by high-ordered spectra (DHS for short) if, whenever $H$ is a graph such that $H^m$ is cospectral with $G^m$ for all $m$, then $H$ is isomorphic to $G$. In this paper we first give formulas for the traces of the power of unicyclic graphs, and then provide some high-ordered cospectral invariants of unicyclic graphs. We prove that a class of unicyclic graphs with cospectral mates is DHS, and give two examples of infinitely many pairs of cospectral unicyclic graphs but with different high-ordered spectra.

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arXiv2022

The trace and Estrada index of uniform hypergraphs with cut vertices

Let $\mathcal{H}$ be an $m$-uniform hypergraph, and let $\mathcal{A}(\mathcal{H})$ be the adjacency tensor of $\mathcal{H}$ which can be viewed as a system of homogeneous polynomials of degree $m-1$. Morozov and Shakirov generalized the traces of linear systems to nonlinear homogeneous polynomial systems and obtained explicit formulas for multidimensional resultants. Sun, Zhou and Bu introduced the Estrada index of uniform hypergraphs which is closely related to the traces of their adjacency tensors. In this paper we give formulas for the traces of $\mathcal{A}(\mathcal{H})$ when $\mathcal{H}$ contains cut vertices, and obtain results on the traces and Estrada index when $\mathcal{H}$ is perturbed under local changes. We prove that among all hypertrees with fixed number of edges, the hyperpath is the unique one with minimum Estrada index and the hyperstar is the unique one with maximum Estrada index.

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arXiv2020

A spectral method to incidence balance of oriented hypergraphs and induced signed hypergraphs

An oriented hypergraph is a hypergraph together with an incidence orientation such that each edge-vertex incidence is given a label of $+1$ or $-1$. An oriented hypergraph is called incidence balanced if there exists a bipartition of the vertex set such that every edge intersects one part of the bipartition in positively incident vertices with the edge and other part in negatively incident vertices with the edge. In this paper, we investigate the incidence balance of oriented hypergraphs and induced signed hypergraphs by means of the eigenvalues of associated matrices and tensors, and provide a spectral method to characterize the incidence balanced oriented hypergraphs and their induced signed hypergraphs.

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arXiv2020

On the largest eigenvalue of a mixed graph with partial orientation

Let $G$ be a connected graph and let $T$ be a spanning tree of $G$. A partial orientation $σ$ of $G$ respect to $T$ is an orientation of the edges of $G$ except those edges of $T$, the resulting graph associated with which is denoted by $G_T^σ$. In this paper we prove that there exists a partial orientation $σ$ of $G$ respect to $T$ such that the largest eigenvalue of the Hermitian adjacency matrix of $G_T^σ$ is at most the largest absolute value of the roots of the matching polynomial of $G$.

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arXiv2020

Minimal non-odd-transversal hypergraphs and minimal non-odd-bipartite hypergraphs

Among all uniform hypergraphs with even uniformity, the odd-transversal or odd-bipartite hypergraphs are more close to bipartite simple graphs from the viewpoint of both structure and spectrum. A hypergraph is called minimal non-odd-transversal if it is non-odd-transversal but deleting any edge results in an odd-transversal hypergraph. In this paper we give an equivalent characterization of the minimal non-odd-transversal hypergraphs by the degrees and the rank of its incidence matrix over $\mathbb{Z}_2$. If a minimal non-odd-transversal hypergraph is uniform, then it has even uniformity, and hence is minimal non-odd-bipartite. We characterize $2$-regular uniform minimal non-odd-bipartite hypergraphs, and give some examples of $d$-regular uniform hypergraphs which are minimal non-odd-bipartite. Finally we give upper bounds for the least H-eigenvalue of the adjacency tensor of minimal non-odd-bipartite hypergraphs.

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arXiv2020

The stabilizing index and cyclic index of coalescence and Cartesian product of uniform hypergraphs

Let $G$ be connected uniform hypergraph and let $\mathcal{A}(G)$ be the adjacency tensor of $G$. The stabilizing index of $G$ is the number of eigenvectors of $\mathcal{A}(G)$ associated with the spectral radius, and the cyclic index of $G$ is the number of eigenvalues of $\mathcal{A}(G)$ with modulus equal to the spectral radius. Let $G_1 \odot G_2$ and $G_1 \Box G_2$ be the coalescence and Cartesian product of connected $m$-uniform hypergraphs $G_1$ and $G_2$ respectively. In this paper, we give explicit formulas for the the stabilizing indices and cyclic indices of $G_1 \odot G_2$ and $G_1 \Box G_2$ in terms of those of $G_1$ and $G_2$ or the invariant divisors of their incidence matrices over $\mathbb{Z}_m$, respectively.

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arXiv2020

The least H-eigenvalue of adjacency tensor of hypergraphs with cut vertices

Let $G$ be a connected hypergraph with even uniformity, which contains cut vertices. Then $G$ is the coalescence of two nontrivial connected sub-hypergraphs (called branches) at a cut vertex. Let $\mathcal{A}(G)$ be the adjacency tensor of $G$. The least H-eigenvalue of $\mathcal{A}(G)$ refers to the least real eigenvalue of $\mathcal{A}(G)$ associated with a real eigenvector. In this paper we obtain a perturbation result on the least H-eigenvalue of $\mathcal{A}(G)$ when a branch of $G$ attached at one vertex is relocated to another vertex, and characterize the unique hypergraph whose least H-eigenvalue attains the minimum among all hypergraphs in a certain class of hypergraphs which contain a fixed connected hypergraph.

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arXiv2020

The linear unicyclic hypergraph with the second or third largest spectral radius

The spectral radius of a uniform hypergraph is defined to be that of the adjacency tensor the hypergraph. It is known that the unique unicyclic hypergraph with the largest spectral radius is a nonlinear hypergraph, and the unique linear unicyclic hypergraph with the largest spectral radius is a power hypergraph. In this paper we determine the the unique linear unicyclic hypergraph with the second or third largest spectral radius, where the former hypergraph is a power hypergraph and the latter hypergraph is a non-power hypergraph.

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arXiv2019

The isomorphism problem of trees from the viewpoint of Terwilliger algebras

Let $Γ^{(x_0)}$ be a finite rooted tree, for which $Γ$ is the underlying tree and $x_0$ the root. Let $T$ be the Terwilliger algebra of $Γ$ with respect to $x_0$. We study the structure of the principal $T$-module. As a result, it is shown that $T$ recognizes the isomorphism class of $Γ^{(x_0)}$.

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arXiv2019

The least H-eigenvalue of signless Laplacian of non-odd-bipartite hypergraphs

Let $G$ be a connected non-odd-bipartite hypergraph with even uniformity. The least H-eigenvalue of the signless Laplacian tensor of $G$ is simply called the least eigenvalue of $G$ and the corresponding H-eigenvectors are called the first eigenvectors of $G$. In this paper we give some numerical and structural properties about the first eigenvectors of $G$ which contains an odd-bipartite branch, and investigate how the least eigenvalue of $G$ changes when an odd-bipartite branch attached at one vertex is relocated to another vertex. We characterize the hypergraph(s) whose least eigenvalue attains the minimum among a certain class of hypergraphs which contain a fixed non-odd-bipartite connected hypergraph. Finally we present some upper bounds of the least eigenvalue and prove that zero is the least limit point of the least eigenvalues of connected non-odd-bipartite hypergraphs.

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arXiv2019

Eigenvectors of Z-tensors associated with least H-eigenvalue with application to hypergraphs

Unlike an irreducible $Z$-matrices, a weakly irreducible $Z$-tensor $\mathcal{A}$ can have more than one eigenvector associated with the least H-eigenvalue. We show that there are finitely many eigenvectors of $\mathcal{A}$ associated with the least H-eigenvalue. If $\mathcal{A}$ is further combinatorial symmetric, the number of such eigenvectors can be obtained explicitly by the Smith normal form of the incidence matrix of $\mathcal{A}$. When applying to a connected uniform hypergraph $G$, we prove that the number of Laplacian eigenvectors of $G$ associated with the zero eigenvalue is equal to the the number of adjacency eigenvectors of $G$ associated with the spectral radius, which is also equal to the number of signless Laplacian eigenvectors of $G$ associated with the zero eigenvalue if zero is an signless Laplacian eigenvalue.

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