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Lihua You

Publications and source records attributed to Lihua You.

At least 37 records · Page 2Linked to original sources
arXiv2024

Connected graphs with large multiplicity of $-1$ in the spectrum of the eccentricity matrix

The eccentricity matrix of a simple connected graph is obtained from the distance matrix by only keeping the largest distances for each row and each column, whereas the remaining entries become zero. This matrix is also called the anti-adjacency matrix, since the adjacency matrix can also be obtained from the distance matrix but this time by keeping only the entries equal to $1$. It is known that, for $λ\not\in \{-1,0\}$ and a fixed $i\in \mathbb{N}$, there is only a finite number of graphs with $n$ vertices having $λ$ as an eigenvalue of multiplicity $n-i$ on the spectrum of the adjacency matrix. This phenomenon motivates researchers to consider the graphs has a large multiplicity of an eigenvalue in the spectrum of the eccentricity matrix, for example, the eigenvalue $-2$ [X. Gao, Z. Stanić, J.F. Wang, Grahps with large multiplicity of $-2$ in the spectrum of the eccentricity matrix, Discrete Mathematics, 347 (2024) 114038]. In this paper, we characterize the connected graphs with $n$ vertices having $-1$ as an eigenvalue of multiplicity $n-i$ $(i\leq5)$ in the spectrum of the eccentricity matrix. Our results also become meaningful in the framework of the median eigenvalue problem [B. Mohar, Median eigenvalues and the HOMO-LUMO index of graphs, Journal of Combinatorial Theory Series B, 112 (2015) 78-92].

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arXiv2024

On determinants of tournaments and $\mathcal{D}_k$

Let $T$ be a tournament with $n$ vertices $v_1,\ldots,v_n$. The skew-adjacency matrix of $T$ is the $n\times n$ zero-diagonal matrix $S_T = [s_{ij}]$ in which $s_{ij}=-s_{ji}=1$ if $ v_i $ dominates $ v_j $. We define the determinant $\det(T)$ of $ T $ as the determinant of $ S_T $. It is well-known that $\det(T)=0$ if $n$ is odd and $\det(T)$ is the square of an odd integer if $n$ is even. Let $\mathcal{D}_k$ be the set of tournaments whose all subtournaments have determinant at most $ k^{2} $, where $k$ is a positive odd integer. The necessary and sufficient condition for $T\in \mathcal{D}_1$ or $T\in \mathcal{D}_3$ has been characterized in $2023$. In this paper, we characterize the set $\mathcal{D}_5$, obtain some properties of $\mathcal{D}_k$. Moreover, for any positive odd integer $k$, we give a construction of a tournament $T$ satisfying that $\det(T)=k^2$, and $T\in \mathcal{D}_k\backslash\mathcal{D}_{k-2}$ if $k\geq 3$, which implies $\mathcal{D}_k\backslash\mathcal{D}_{k-2}$ is not an empty set for $k\geq 3$.

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arXiv2024

Deep Spectral Meshes: Multi-Frequency Facial Mesh Processing with Graph Neural Networks

With the rising popularity of virtual worlds, the importance of data-driven parametric models of 3D meshes has grown rapidly. Numerous applications, such as computer vision, procedural generation, and mesh editing, vastly rely on these models. However, current approaches do not allow for independent editing of deformations at different frequency levels. They also do not benefit from representing deformations at different frequencies with dedicated representations, which would better expose their properties and improve the generated meshes' geometric and perceptual quality. In this work, spectral meshes are introduced as a method to decompose mesh deformations into low-frequency and high-frequency deformations. These features of low- and high-frequency deformations are used for representation learning with graph convolutional networks. A parametric model for 3D facial mesh synthesis is built upon the proposed framework, exposing user parameters that control disentangled high- and low-frequency deformations. Independent control of deformations at different frequencies and generation of plausible synthetic examples are mutually exclusive objectives. A Conditioning Factor is introduced to leverage these objectives. Our model takes further advantage of spectral partitioning by representing different frequency levels with disparate, more suitable representations. Low frequencies are represented with standardised Euclidean coordinates, and high frequencies with a normalised deformation representation (DR). This paper investigates applications of our proposed approach in mesh reconstruction, mesh interpolation, and multi-frequency editing. It is demonstrated that our method improves the overall quality of generated meshes on most datasets when considering both the $L_1$ norm and perceptual Dihedral Angle Mesh Error (DAME) metrics.

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arXiv2023

Spectral extrema of graphs: Forbidden star-path forests

A path of order $n$ is denoted by $P_n$, and a star of order $n$ is denoted by $S_{n-1}$. A star-path forest is a forest whose connected components are paths and stars. In this paper we determine the maximum spectral radius of graphs that contain no copy of $kS_{\ell-1}\cup P_{\ell}$, $k_1S_{2\ell -1}\cup k_2P_{2\ell}$ or $kS_{4}\cup 2P_{5}$ for $n$ appropriately large.

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arXiv2022

Extremal Kirchhoff index in polycyclic chains

The Kirchhoff index of graphs, introduced by Klein and Randić in 1993, has been known useful in the study of computer science, complex network and quantum chemistry. The Kirchhoff index of a graph $G$ is defined as $Kf(G)=\sum\limits_{\{u,v\}\subseteq V(G)}Ω_{G}(u,v)$, where $Ω_{G}(u,v)$ denotes the resistance distance between $u$ and $v$ in $G$. In this paper, we determine the maximum (resp. minimum) $k$-polycyclic chains with respect to Kirchhoff index for $k\geq 5$, which extends the results of Yang and Klein [Comparison theorems on resistance distances and Kirchhoff indices of $S,T$-isomers, Discrete Appl. Math. 175 (2014) 87-93], Yang and Sun [Minimal hexagonal chains with respect to the Kirchhoff index, Discrete Math. 345 (2022) 113099], Sun and Yang [Extremal pentagonal chains with respect to the Kirchhoff index, Appl. Math. Comput. 437 (2023) 127534] and Ma [Extremal octagonal chains with respect to the Kirchhoff index, arXiv: 2209.10264].

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arXiv2022

Regular graphs with a complete bipartite graph as a star complement

Let $G$ be a graph of order $n$ and $μ$ be an adjacency eigenvalue of $G$ with multiplicity $k\geq 1$. A star complement $H$ for $μ$ in $G$ is an induced subgraph of $G$ of order $n-k$ with no eigenvalue $μ$, and the vertex subset $X=V(G-H)$ is called a star set for $μ$ in $G$. The study of star complements and star sets provides a strong link between graph structure and linear algebra. In this paper, we study the regular graphs with $K_{t,s}\ (s\geq t\geq 2)$ as a star complement for an eigenvalue $μ$, especially, characterize the case of $t=3$ completely, obtain some properties when $t=s$, and propose some problems for further study.

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arXiv2022

The maximum spectral radius of graphs of given size with forbidden subgraph

Let $G$ be a graph of size $m$ and $ρ(G)$ be the spectral radius of its adjacency matrix. A graph is said to be $F$-free if it does not contain a subgraph isomorphic to $F$. In this paper, we prove that if $G$ is a $K_{2,r+1}$-free non-star graph with $m\geq (4r+2)^2+1$, then $ρ(G)\leq ρ(S_m^1)$, with equality if and only if $G\cong S_m^1$. Recently, Li, Sun and Wei showed that for any $θ_{1,2,3}$-free graph of size $m\geq 8$, $ρ(G)\leq \frac{1+\sqrt{4m-3}}{2}$, with equality if and only if $G\cong S_{\frac{m+3}{2},2}$. However, this bound is not attainable when $m$ is even. We proved that if $G$ is $θ_{1,2,3}$-free and $G\ncong S_{\frac{m+3}{2},2}$ with $m\geq 22$, then $ρ(G)\leq ρ(F_{m,1})$ if $m$ is even, with equality if and only if $G\cong F_{m,1}$, and $ρ(G)\leq ρ(F_{m,2})$ if $m$ is odd, with equality if and only if $G\cong F_{m,2}$.

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arXiv2022

The Sombor index and coindex of two-trees

The Sombor index of a graph $G$, introduced by Ivan Gutman, is defined as the sum of the weights $\sqrt{d_G(u)^2+d_G(v)^2}$ of all edges $uv$ of $G$, where $d_G(u)$ denotes the degree of vertex $u$ in $G$. The Sombor coindex is recently defined as $\bar{SO}(G)=\sum \limits_{uv\notin E(G)}\sqrt{d_G(u)^2+d_G(v)^2}$. In this paper, the maximum and second maximum Sombor index, the minimum and second minimum Sombor coindex in two-trees are determined.

math.CO↗
arXiv2021

On the first three minimum Mostar indices of tree-like phenylenes

Let $G =(V_{G}, E_{G})$ be a simple connected graph with its vertex set $V_{G}$ and edge set $E_{G}$. The Mostar index $Mo(G)$ was defined as $Mo(G)=\sum\limits_{e=uv\in E(G)}|n_{u}-n_{v}|$, where $n_{u}$ (resp., $n_{v}$) is the number of vertices whose distance to vertex $u$ (resp., $v$) is smaller than the distance to vertex $v$ (resp., $u$). In this study, we determine the first three minimum Mostar indices of tree-like phenylenes and characterize all the tree-like phenylenes attaining these values. At last, we give some numerical examples and discussion.

math.CO↗
arXiv2021

Spectral properties of $p$-Sombor matrices and beyond

Let $G=(V(G),E(G))$ be a simple graph with vertex set $V(G)=\{v_{1},v_{2},\cdots, v_{n}\}$ and edge set $E(G)$. The $p$-Sombor matrix $\mathcal{S}_{p}(G)$ of $G$ is the square matrix of order $n$ whose $(i,j)$-entry is equal to $((d_{i})^{p}+(d_{j})^{p})^{\frac{1}{p}}$ if $v_{i}\sim v_{j}$, and 0 otherwise, where $d_{i}$ denotes the degree of vertex $v_{i}$ in $G$. In this paper, we study the relationship between $p$-Sombor index $SO_{p}(G)$ and $p$-Sombor matrix $\mathcal{S}_{p}(G)$ by the $k$-th spectral moment $N_{k}$ and the spectral radius of $\mathcal{S}_{p}(G)$. Then we obtain some bounds of $p$-Sombor Laplacian eigenvalues, $p$-Sombor spectral radius, $p$-Sombor spectral spread, $p$-Sombor energy and $p$-Sombor Estrada index. We also investigate the Nordhaus-Gaddum-type results for $p$-Sombor spectral radius and energy. At last, we give the regression model for boiling point and some other invariants.

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arXiv2021

Ordering chemical graphs by Sombor indices and its applications

Topological indices are a class of numerical invariants that predict certain physical and chemical properties of molecules. Recently, two novel topological indices, named as Sombor index and reduced Sombor index, were introduced by Gutman, defined as $$SO(G)=\sum_{uv\in E(G)}\sqrt{d_{G}^{2}(u)+d_{G}^{2}(v)},$$ $$SO_{red}(G)=\sum_{uv\in E(G)}\sqrt{(d_{G}(u)-1)^{2}+(d_{G}(v)-1)^{2}},$$ where $d_{G}(u)$ denotes the degree of vertex $u$ in $G$. In this paper, our aim is to order the chemical trees, chemical unicyclic graphs, chemical bicyclic graphs and chemical tricyclic graphs with respect to Sombor index and reduced Sombor index. We determine the first fourteen minimum chemical trees, the first four minimum chemical unicyclic graphs, the first three minimum chemical bicyclic graphs, the first seven minimum chemical tricyclic graphs. At last, we consider the applications of reduced Sombor index to octane isomers.

math.CO↗
arXiv2021

The expected values of Sombor indices in random hexagonal chains, phenylene chains and Sombor indices of some chemical graphs

Hexagonal chains are a special class of catacondensed benzenoid system and phenylene chains are a class of polycyclic aromatic compounds. Recently, A family of Sombor indices was introduced by Gutman in the chemical graph theory. It had been examined that these indices may be successfully applied on modeling thermodynamic properties of compounds. In this paper, we study the expected values of the Sombor indices in random hexagonal chains, phenylene chains, and consider the Sombor indices of some chemical graphs such as graphene, coronoid systems and carbon nanocones.

math.CO↗
arXiv2017

Some results of strongly primitive tensors

In this paper, we show that an order $m$ dimension 2 tensor is primitive if and only if its majorization matrix is primitive, and then we obtain the characterization of order $m$ dimension 2 strongly primitive tensors and the bound of the strongly primitive degree. Furthermore, we study the properties of strongly primitive tensors with $n\geq 3$, and propose some problems for further research.

math.CO↗
arXiv2016

Spectrum of a class of matrices and its applications

In this paper, we give the spectrum of a matrix by using the quotient matrix, then we apply this result to various matrices associated to a graph and a digraph, including adjacency matrix, (signless) Laplacian matrix, distance matrix, distance (signless) Laplacian matrix, to obtain some known and new results. Moreover, we propose some problems for further research.

math.CO↗
arXiv2016

A Sharp upper bound for the spectral radius of a nonnegative matrix and applications

In this paper, we obtain a sharp upper bound for the spectral radius of a nonnegative matrix. This result is used to present upper bounds for the adjacency spectral radius, the Laplacian spectral radius, the signless Laplacian spectral radius, the distance spectral radius, the distance Laplacian spectral radius, the distance signless Laplacian spectral radius of a graph or a digraph. These results are new or generalize some known results.

math.CO↗
arXiv2016

Distance and distance signless Laplacian spread of connected graphs

For a connected graph $G$ on $n$ vertices, recall that the distance signless Laplacian matrix of $G$ is defined to be $\mathcal{Q}(G)=Tr(G)+\mathcal{D}(G)$, where $\mathcal{D}(G)$ is the distance matrix, $Tr(G)=diag(D_1, D_2, \ldots, D_n)$ and $D_{i}$ is the row sum of $\mathcal{D}(G)$ corresponding to vertex $v_{i}$. Denote by $ρ^{\mathcal{D}}(G),$ $ρ_{min}^{\mathcal{D}}(G)$ the largest eigenvalue and the least eigenvalue of $\mathcal{D}(G)$, respectively. And denote by $q^{\mathcal{D}}(G)$, $q_{min}^{\mathcal{D}}(G)$ the largest eigenvalue and the least eigenvalue of $\mathcal{Q}(G)$, respectively. The distance spread of a graph $G$ is defined as $S_{\mathcal{D}}(G)=ρ^{\mathcal{D}}(G)- ρ_{min}^{\mathcal{D}}(G)$, and the distance signless Laplacian spread of a graph $G$ is defined as $S_{\mathcal{Q}}(G)=q^{\mathcal{D}}(G)-q_{min}^{\mathcal{D}}(G)$. In this paper, we point out an error in the result of Theorem 2.4 in "Distance spectral spread of a graph" [G.L. Yu, et al, Discrete Applied Mathematics. 160 (2012) 2474--2478] and rectify it. As well, we obtain some lower bounds on ddistance signless Laplacian spread of a graph.

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