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Lizhu Sun

Publications and source records attributed to Lizhu Sun.

15 recordsLinked to original sources
arXiv2026

Quadratic form estimations for Hessian matrices of resistance distance and Kirchhoff index of positive-weighted graphs

Let $G^{w}=(V,E,w)$ be a positive-weighted graph with the weight $w(e)>0$ for all $e\in E$. The weighted graph $G^{\widetilde{w}}=(V,E,\widetilde{w})$ is called a hyper-dual number weighted graph, where the weight $\widetilde{w}(e)=w(e)+\Delta w(e)(\varepsilon+\varepsilon^{*})$ is a hyper dual number, $\Delta w(e)$ is a real number, $\varepsilon$ and $\varepsilon^{*}$ are two dual units, $e\in E$. In this paper, we give a representation for the Moore-Penrose inverse of the Laplacian matrix, and calculation formulas for the resistance distance and Kirchhoff index of $G^{\widetilde{w}}$, respectively. We establish quadratic forms of the Hessian matrices for the resistance distance and Kirchhoff index of $G^{w}$ via generalized matrix inverses. We further derive explicit bounds on the eigenvalues of the Hessian matrices for the resistance distance and the Kirchhoff index of $G^{w}$ in terms of graph parameters. We also prove that the Kirchhoff index of a positive-weighted graph with bounded edge weights is strongly convex on its edge weight vector.

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arXiv2025

The high order spectral radius of graphs without long cycles or paths

In 1959, Erd\H{o}s and Gallai established two classic theorems, which determine the maximum number of edges in an $n$-vertex graph with no cycles of length at least $k$, and in an $n$-vertex graph with no paths on $k$ vertices, respectively. Subsequently, generalized and spectral versions of the Erd\H{o}s-Gallai theorems have been investigated. A concept of a high order spectral radius for graphs was introduced in 2023, defined as the spectral radius of a tensor and termed the $t$-clique spectral radius $\rho_t(G)$. In this paper, we establish a high order spectral version of Erd\H{o}s-Gallai theorems by employing the $t$-clique spectral radius, i.e., we determine the extremal graphs that attain the maximum $t$-clique spectral radius in the $n$-vertex graphs with no cycles of length at least $k$ and in the $n$-vertex graphs with no paths on $k$ vertices, respectively.

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arXiv2025

The ordering of hypertrees and unicyclic hypergraphs by the traces of $\mathcal{A}_{\alpha}$-tensor

For a real number $\alpha\in[0,1]$ and a $k$-uniform hypergraph $\mathcal{H}$, $\mathcal{A}_{\alpha}(\mathcal{H})=\alpha\mathcal{D}(\mathcal{H})+(1-\alpha)\mathcal{A}(\mathcal{H})$ is called the $\mathcal{A}_{\alpha}$-tensor of $\mathcal{H}$, where $\mathcal{D}(\mathcal{H})$ and $\mathcal{A}(\mathcal{H})$ are the degree tensor and adjacency tensor of $\mathcal{H}$, respectively. The sum of the $d$-th powers of all eigenvalues of $\mathcal{A}_{\alpha}(\mathcal{H})$ is called the $d$-th order $\mathcal{A}_{\alpha}$-spectral moment of $\mathcal{H}$, which is equal to the $d$-th order trace of $\mathcal{A}_{\alpha}(\mathcal{H})$. In this paper, some hypergraphs are ordered lexicographically by their $\mathcal{A}_{\alpha}$-spectral moments in non-decreasing order. The first, the second, the last and the second last hypergraphs among all $k$-uniform linear unicyclic hypergraphs and hypertrees are characterized, respectively. We give the first and the last hypergraphs among all $k$-uniform linear unicyclic hypergraphs with given grith, and characterize the last hypertree among all $k$-uniform hypertrees with given diameter. Furthermore, we determine some extreme values of the $\mathcal{A}_{\alpha}$-spectral moments for hypertrees and linear unicyclic hypergraphs, respectively.

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arXiv2025

The resistance distance of a dual number weighted graph

For a graph $G=(V,E)$, assigning each edge $e\in E$ a weight of a dual number $w(e)=1+\widehat{a}_{e}\varepsilon$, the weighted graph $G^{w}=(V,E,w)$ is called a dual number weighted graph, where $-\widehat{a}_{e}$ can be regarded as the perturbation of the unit resistor on edge $e$ of $G$. For a connected dual number weighted graph $G^{w}$, we give some expressions and block representations of generalized inverses of the Laplacian matrix of $G^{w}$. And using these results, we derive the explicit formulas of the resistance distance and Kirchhoff index of $G^{w}$. We give the perturbation bounds for the resistance distance and Kirchhoff index of $G$. In particular, when only the edge $e=\{i,j\}$ of $G$ is perturbed, we give the perturbation bounds for the Kirchhoff index and resistance distance between vertices $i$ and $j$ of $G$, respectively.

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arXiv2024

The subgraph eigenvector centrality of graphs

Let $G$ be a connected graph and let $F$ be a connected subgraph of $G$ with a given structure. We consider that the centrality of a vertex $i$ of $G$ is determined by the centrality of other vertices in all subgraphs contain $i$ and isomorphic to $F$. In this paper we propose an $F$-subgraph tensor and an $F$-subgraph eigenvector centrality of $G$. When the graph is $F$-connected, we show that the $F$-subgraph tensor is weakly irreducible, and in this case, the $F$-subgraph eigenvector centrality exists. Specifically, when we choose $F$ to be a path $P_1$ of length $1$(or a complete graph $K_2$), the $F$-eigenvector centrality is eigenvector centrality of $G$. Furthermore, we propose the $(K_2,F)$-subgraph eigenvector centrality of $G$ and prove it always exists when $G$ is connected. Specifically, the $P_2$-subgraph eigenvector centrality and $(K_2,F)$-subgraph eigenvector centrality are studied. Some examples show that the ranking of vertices under them differs from the rankings under several classic centralities. Vertices of a regular graph have the same eigenvector centrality scores. But the $(K_2,K_3)$-subgraph eigenvector centrality can distinguish vertices in a given regular graph.

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arXiv2023

Estrada index and subgraph centrality of hypergraphs via tensors

Uniform hypergraphs have a natural one-to-one correspondence to tensors. In this paper, we investigate the Estrada index and subgraph centrality of an $m$-uniform hypergraph $\mathcal{H}$ via the adjacency tensor. We establish some bounds for the Estrada index and give expressions of the subgraph centrality in terms of graph parameters of the multi-digraphs associated with $\mathcal{H}$. When $\mathcal{H}$ is $2$-uniform, the above Estrada index and subgraph centrality are the Estrada index and subgraph centrality of a graph.

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arXiv2021

High-ordered spectral characterizations of graphs

The spectrum of the $k$-power hypergraph of a graph $G$ is called the $k$-ordered spectrum of $G$.If graphs $G_1$ and $G_2$ have same $k$-ordered spectrum for all positive integer $k\geq2$, $G_1$ and $G_2$ are said to be high-ordered cospectral. If all graphs who are high-ordered cospectral with the graph $G$ are isomorphic to $G$, we say that $G$ is determined by the high-ordered spectrum.In this paper, we use the high-ordered spectrum of graphs to study graph isomorphism and show that all Smith's graphs are determined by the high-ordered spectrum.And we give infinitely many pairs of trees with same spectrum but different high-ordered spectrum by high-ordered cospectral invariants of trees,it means that we can determine that these cospectral trees are not isomorphism by the high-ordered spectrum.

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arXiv2019

The geometry connectivity of hypergraphs

Let $\mathcal{G}$ be a $k$-uniform hypergraph, $\mathcal{L}_{\mathcal{G}}$ be its Laplacian tensor. And $β( \mathcal{G})$ denotes the maximum number of linearly independent nonnegative eigenvectors of $\mathcal{L}_{\mathcal{G}}$ corresponding to the eigenvalue $0$. In this paper, $β( \mathcal{G})$ is called the geometry connectivity of $\mathcal{G}$. We show that the number of connected components of $\mathcal{G}$ equals the geometry connectivity $β( \mathcal{G})$.

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arXiv2019

Bounds on the $α$-distance spectrum of graphs

For a simple, undirected and connected graph $G$, $D_α(G) = αTr(G) + (1-α) D(G)$ is called the $α$-distance matrix of $G$, where $α\in [0,1]$, $D(G)$ is the distance matrix of $G$, and $Tr(G)$ is the vertex transmission diagonal matrix of $G$. Recently, the $α$-distance energy of $G$ was defined based on the spectra of $D_α(G)$. In this paper, we define the $α$-distance Estrada index of $G$ in terms of the eigenvalues of $D_α(G)$. And we give some bounds on the spectral radius of $D_α(G)$, $α$-distance energy and $α$-distance Estrada index of $G$.

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arXiv2016

Spectral properties of general hypergraphs

In this paper, we investigate spectral properties of the adjacency tensor, Laplacian tensor and signless Laplacian tensor of general hypergraphs (including uniform and non-uniform hypergraphs). We obtain some bounds for the spectral radius of general hypergraphs in terms of vertex degrees, and give spectral characterizations of odd-bipartite hypergraphs.

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arXiv2015

On the spectral radius of nonregular uniform hypergraphs

Let $G$ be a connected uniform hypergraphs with maximum degree $Δ$, spectral radius $λ$ and minimum H-eigenvalue $μ$. In this paper, we give some lower bounds for $Δ-λ$, which extend the result of [S.M. Cioabă, D.A. Gregory, V. Nikiforov, Extreme eigenvalues of nonregular graphs, J. Combin. Theory, Ser. B 97 (2007) 483-486] to hypergraphs. Applying these bounds, we also obtain a lower bound for $Δ+μ$.

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arXiv2014

Minimum (maximum) rank of tensors and the sign nonsingular tensors

In this paper, we define the minimum (maximum) rank, term rank and the sign nonsingular of tensors. The sufficiency and necessity for the minimum rank of a real tensor to be $1$ is given. And we show that the maximum rank of a tensor is not less than the term rank. We also prove that the minimum rank of a sign nonsingular tensor is not less than the dimension of it. And we get some characterizations of a tensor having sign left or sign right inverses.

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arXiv2014

Some results on the generalized inverse of tensors and idempotent tensors

Let $\mathcal{A}$ be an order $t$ dimension $m\times n\times \cdots \times n$ tensor over complex field. In this paper, we study some {generalized inverses} of $\mathcal{A}$, the {$k$-T-idempotent tensors} and the idempotent tensors based on the general tensor product. Using the tensor generalized inverse, some solutions of the equation $\mathcal{A}\cdot x^{t-1}=b$ are given, where $x$ and $b$ are dimension $n$ and $m$ vectors, respectively. The {generalized inverses} of some block tensors, the eigenvalues of {$k$-T-idempotent tensors} and idempotent tensors are given. And the relation between the generalized inverses of tensors and the $k$-T-idempotent tensors is also showed.

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arXiv2014

Some spectral properties of uniform hypergraphs

For a $k$-uniform hypergraph $H$, we obtain some trace formulas for the Laplacian tensor of $H$, which imply that $\sum_{i=1}^nd_i^s$ ($s=1,\ldots,k$) is determined by the Laplacian spectrum of $H$, where $d_1,\ldots,d_n$ is the degree sequence of $H$. Using trace formulas for the Laplacian tensor, we obtain expressions for some coefficients of the Laplacian polynomial of a regular hypergraph. We give some spectral characterizations of odd-bipartite hypergraphs, and give a partial answer to a question posed by Shao et al \cite{ShaoShanWu}. We also give some spectral properties of power hypergraphs, and show that a conjecture posed by Hu et al \cite{HuQiShao} holds under certain conditons.

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