Romain Boulet
A centipede is a graph obtained by appending a pendant vertex to each vertex of degree 2 of a path. In this paper we prove that the centipede is determined by its Laplacian spectrum.
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Let $F_{p,q}$ be the generalized friendship graph $K_1\bigvee (pK_q)$ on $pq+1$ vertices obtained by joining a vertex to all vertices of $p$ disjoint copies of the complete graph $K_q$ on $q$ vertices. In this paper, we prove that $F_{p,q}$ is determined by its normalized Laplacian spectrum if and only if $q\ge 2$, or $q=1$ and $p\le 2$.
In this article, various aspects of Laplacian spectra of power graphs of finite cyclic, dicyclic and finite $p$-groups are studied. Algebraic connectivity of power graphs of the above groups are considered and determined completely for that of finite $p$-groups. Further, the multiplicity of Laplacian spectral radius of power graphs of the above groups are studied and determined completely for that of dicyclic and finite $p$-groups. The equality of the vertex connectivity and the algebraic connectivity is characterized for power graphs of all of the above groups. Orders of dicyclic groups, for …
In the past decades, graphs that are determined by their spectrum have received more attention, since they have been applied to several fields, such as randomized algorithms, combinatorial optimization problems and machine learning. An important part of spectral graph theory is devoted to determining whether given graphs or classes of graphs are determined by their spectra or not. So, finding and introducing any class of graphs which are determined by their spectra can be an interesting and important problem. A graph is said to be DQS if there is no other non-isomorphic graph with the same sig…
On digraphs determined by their singular values
本文研究了有向路径、有向环和星形图的奇异值，证明了有向路径、有向环和定向星形图由拉普拉斯和无符号拉普拉斯奇异值确定，但不通过邻接矩阵奇异值。
{Signless Laplacian determinations of some graphs with independent edges}% {Let $G$ be a simple undirected graph. Then the signless Laplacian matrix of $G$ is defined as $D_G + A_G$ in which $D_G$ and $A_G$ denote the degree matrix and the adjacency matrix of $G$, respectively. The graph $G$ is said to be determined by its signless Laplacian spectrum ({\rm DQS}, for short), if any graph having the same signless Laplacian spectrum as $G$ is isomorphic to $G$. We show that $G\sqcup rK_2$ is determined by its signless Laplacian spectra under certain conditions, where $r$ and $K_2$ denote a natura…
The main goal of the paper is to characterize new classes of multicone graphs which are determined by both adjacency and Laplacian spectra. A multicone graph is defined to be the join of a clique and a regular graph. A wheel graph obtained from the join of a complete graph on a vertex with a cycle. A question about when wheel graphs are determined by their adjacency spectra is still unsolved. So, any indication about the determinations of these graphs with respect to their adjacency spectra can be an interesting and important problem. In [Y. Zhang, X. Liu, and X. Yong: Which wheel graphs are d…
The graph entropy describes the structural information of graph. Motivated by the definition of graph entropy in general graphs, the graph entropy of hypergraphs based on Laplacian degree are defined. Some results on graph entropy of simple graphs are extended to k-uniform hypergraphs. Using an edge-moving operation, the maximum and minimum graph entropy based on Laplacian degrees are determined in k-uniform hypertrees, unicyclic k-uniform hypergraphs, bicyclic k-uniform hypergraphs and k-uniform chemical hypertrees, respectively, and the corresponding extremal graphs are determined.
On Laplacian and Signless Laplacian Permanental Polynomials of Some Well-known Graphs
本文研究了若干著名星状图的拉普拉斯与无符号拉普拉斯积和多项式，确定了这些图可由单一或联合多项式进行刻画的条件，并提出了相关的计算技术。
Graphs with few distinct eigenvalues have been investigated extensively. In this paper, we focus on another relevant topic: characterizing graphs with some eigenvalue of large multiplicity. Specifically, the normalized Laplacian matrix of a graph is considered here. Let $ρ_{n-1}(G)$ and $ν(G)$ be the second least normalized Laplacian eigenvalue and the independence number of a graph $G$, respectively. As the main conclusions, two families of $n$-vertex connected graphs with some normalized Laplacian eigenvalue of multiplicity $n-3$ are determined: graphs with $ρ_{n-1}(G)=-1$ and graphs with $ρ…
A signless Laplacian eigenvalue of a graph $G$ is called a main signless Laplacian eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, all connected bicyclic graphs with exactly two main eigenvalues are determined.
We introduce the matrix sums that represent a discrete analog of the matrix models with quartic potential. The probability space is given by the set of all simple n-vertex graphs with the Gibbs weight determined by the graph Laplacian. We study the large-n limit of the free energy per site and show that it is determined by the number of connected acyclic diagrams on the set of two-valent vertices.
Let $G$ be a graph with $n$ vertices, and let $L(G)$ and $Q(G)$ be the Laplacian matrix and signless Laplacian matrix of $G$, respectively. The polynomial $π(L(G);x)={\rm per}(xI-L(G))$ (resp. $π(Q(G);x)={\rm per}(xI-Q(G))$) is called {\em Laplacian permanental polynomial} (resp. {\em signless Laplacian permanental polynomial}) of $G$. In this paper, we show that two classes of bicyclic graphs are determined by their (signless) Laplacian permanental polynomials.
The study and calculation of spectrum of networks can be used to describe networks structure and quantify analysis of networks performance. The fractal Möbius octagonal networks, denoted by $Q_n$, is derived from the inverse identification of the opposite lateral edges of fractal linear octagonal networks. In this paper, the normalized Laplacian spectrum of $Q_n$ is determined by two matrices $\mathcal {L}_A$ and $\mathcal {L}_S$. As an important application of our results, some topological indices (multiplicative degree-Kirchhoff index, the number of spanning trees) formulas of $Q_n$ are obta…
The trees, respectively unicyclic graphs, on $n$ vertices with the smallest Laplacian permanent are studied. In this paper, by edge-grafting transformations, the $n$-vertex trees of given bipartition having the second and third smallest Laplacian permanent are identified. Similarly, the $n$-vertex bipartite unicyclic graphs of given bipartition having the first, second and third smallest Laplacian permanent are characterized. Consequently, the $n$-vertex bipartite unicyclic graphs with the first, second and third smallest Laplacian permanent are determined.
Almost complete graphs determined by Laplacian hook immanantal polynomials
研究\(n>7\)且\(n\neq 2k - 1\)时，\(\mathscr{G}_n\)中简单图由拉普拉斯钩子不变多项式\(\Phi_k(L(G),x)\)确定的问题，通过恢复相关参数并基于补图分类比较系数来证明，\(n = 2k - 1\)情况未解决。
Let $λ_{2}(G)$ be the second smallest normalized Laplacian eigenvalue of a graph $G$. In this paper, we determine all unicyclic graphs of order $n\geq21$ with $λ_{2}(G)\geq 1-\frac{\sqrt{6}}{3}$. Moreover, the unicyclic graphs with $λ_{2}(G)=1-\frac{\sqrt{6}}{3}$ are also determined.
The main aim of this study is to characterize new classes of multicone graphs which are determined by their signless Laplacian spectra, their Laplacian spectra and their adjacency spectra. A multicone graph is defined to be the join of a clique and a regular graph. Let P and Kw denote the Petersen graph and a complete graph on w vertices, respectively. In this paper, we show that multicone graphs Kw+P are determined by their signless Laplacian spectra, their Laplacian spectra and their adjacency spectra. Keywords: DS graph; Multicone graph; Signless Laplacian spectrum; Petersen graph.
A tree is called double starlike if it has exactly two vertices of degree greater than two. Let $H(p,n,q)$ denote the double starlike tree obtained by attaching $p$ pendant vertices to one pendant vertex of the path $P_n$ and $q$ pendant vertices to the other pendant vertex of $P_n$. In this paper, we prove that $H(p,n,q)$ is determined by its Laplacian spectrum.
Let $\mathcal{G}(n,k)$ be the set of connected graphs of order $n$ with one of the Laplacian eigenvalue having multiplicity $k$. It is well known that $\mathcal{G}(n,n-1)=\{K_n\}$. The graphs of $\mathcal{G}(n,n-2)$ are determined by Das, and the graphs of $\mathcal{G}(n,n-3)$ with four distinct Laplacian eigenvalues are determined by Mohammadian et al. In this paper, we determine the graphs of $\mathcal{G}(n,n-3)$ with three distinct Laplacian eigenvalues, and then the full characterization of the graphs in $\mathcal{G}(n,n-3)$ is completed.
We introduce a zeta function of digraphs that determines, and is determined by, the spectra of all linear combinations of the adjacency matrix, its transpose, the out-degree matrix, and the in-degree matrix. In particular, zeta-equivalence of graphs encompasses simultaneous cospectrality with respect to the adjacency, the Laplacian, the signless Laplacian, and the normalized Laplacian matrix, respectively. In addition, we express zeta-equivalence in terms of Markov chains and in terms of invasions where each edge is replaced by a fixed digraph. We finish with a method for constructing zeta-equ…
We investigate the structure of trees that have greatest maximum eigenvalue among all trees with a given degree sequence. We show that in such an extremal tree the degree sequence is non-increasing with respect to an ordering of the vertices that is obtained by breadth-first search. This structure is uniquely determined up to isomorphism. We also show that the maximum eigenvalue in such classes of trees is strictly monotone with respect to majorization.
Let Ln denote linear octagonal-quadrilateral networks. In this paper, we aim to firstly investigate the Laplacian spectrum on the basis of Laplacian polynomial of Ln. Then, by applying the relationship between the coefficients and roots of the polynomials, the Kirchhoff index and the complexity are determined.
A rose graph is a graph consisting of cycles that all meet in one vertex. We show that except for two specific examples, these rose graphs are determined by the Laplacian spectrum, thus proving a conjecture posed by Lui and Huang [F.J. Liu and Q.X. Huang, Laplacian spectral characterization of 3-rose graphs, Linear Algebra Appl. 439 (2013), 2914--2920]. We also show that if two rose graphs have a so-called universal Laplacian matrix with the same spectrum, then they must be isomorphic. In memory of Horst Sachs (1927-2016), we show the specific case of the latter result for the adjacency matrix…
Let $G$ be a connected simple graph of order $n$. Let $ρ_1(G)\geq ρ_2(G)\geq \cdots \geq ρ_{n-1}(G)> ρ_n(G)=0$ be the eigenvalues of the normalized Laplacian matrix $\mathcal{L}(G)$ of $G$. Denote by $m(ρ_i)$ the multiplicity of the normalized Laplacian eigenvalue $ρ_i$. Let $ν(G)$ be the independence number of $G$. In this paper, we give a full characterization of graphs with some normalized Laplacian eigenvalue of multiplicity $n-3$, which answers a remaining problem in [S. Sun, K.C. Das, On the multiplicities of normalized Laplacian eigenvalues of graphs, Linear Algebra Appl. 609 (2021) 365…
A connected graph $G$ is a cactus if any two of its cycles have at most one common vertex. Let $\ell_n^m$ be the set of cacti on $n$ vertices with matching number $m.$ S.C. Li and M.J. Zhang determined the unique graph with the maximum signless Laplacian spectral radius among all cacti in $\ell_n^m$ with $n=2m$. In this paper, we characterize the case $n\geq 2m+1$. This confirms the conjecture of Li and Zhang(S.C. Li, M.J. Zhang, On the signless Laplacian index of cacti with a given number of pendant vetices, Linear Algebra Appl. 436, 2012, 4400--4411). Further, we characterize the unique grap…
The $k$-parallel subdivision graph $S_k(G)$ is generated from $G$ which each edge of $G$ is replaced by $k$ parallel paths of length 2. The $2k$-parallel subdivision graph $S_{2k}(G)$ is constructed from $G$ which each edge of $G$ is replaced by $k$ parallel paths of length 3. In this paper, the normalized Laplacian spectra of $S_k(G)$ and $S_{2k}(G)$ are given. They turn out that the multiplicities of the corresponding eigenvalues are only determined by $k$. As applications, the expected hitting time, the expected commute time and any two-points resistance distance between vertices $i$ and $j…
Let $P_n$ and $C_n$ denote the path and cycle on $n$ vertices respectively. The dumbbell graph, denoted by $D_{p,k,q}$, is the graph obtained from two cycles $C_p$, $C_q$ and a path $P_{k+2}$ by identifying each pendant vertex of $P_{k+2}$ with a vertex of a cycle respectively. The theta graph, denoted by $Θ_{r,s,t}$, is the graph formed by joining two given vertices via three disjoint paths $P_{r}$, $P_{s}$ and $P_{t}$ respectively. In this paper, we prove that all dumbbell graphs as well as theta graphs are determined by their Laplacian spectra.
A signed graph product and its Application
该研究定义了一种新的带符号图乘积，推导了其及对应无符号图乘积的各类谱特性，确定了其为整谱图的条件，并构造出同谱与非同谱等能的带符号图序列。
A graph $G$ is said to be determined by the spectrum of its Laplacian matrix (DLS) if every graph with the same spectrum is isomorphic to $G$. van Dam and Haemers (2003) conjectured that almost all graphs have this property, but that is known to be the case only for a very few families. In some recent papers it is proved that the friendship graphs and starlike trees are DLS. If a friendship graph and a starlike tree are joined by merging their vertices of degree greater than 2, then the resulting graph is called a path-friendship graph. In this paper, it is proved that the path-friendship grap…
Let $B_n$ be a linear polyomino chain with $n$ squares. Let $B_n^2$ be the graph obtained by the strong prism of a linear polyomino chain with $n$ squares, i.e. the strong product of $K_2$ and $B_n$. In this paper, explicit expressions for degree-Kirchhoff index and number of spanning trees of $B^2_n$ are determined, respectively. Furthermore, it is interesting to find that the degree-Kirchhoff index of $B^2_n$ is almost one eighth of its Gutman index.
Let $m_GI$ denote the number of Laplacian eigenvalues of a graph $G$ in an interval $I$ and let $α(G)$ denote the independence number of $G$. In this paper, we determine the classes of graphs that satisfy the condition $m_G[0,n-α(G)]=α(G)$ when $α(G)= 2$ and $α(G)= n-2$, where $n$ is the order of $G$. When $α(G)=2$, $G \cong K_1 \nabla K_{n-m} \nabla K_{m-1}$ for some $m \geq 2$. When $α(G)=n-2$, there are two types of graphs $B(p,q,r)$ and $B'(p,q,r)$ of order $n=p+q+r+2$, which we call the binary star graphs. Also, we show that the binary star graphs with $p=r$ are determined by their Laplac…
For a simple digraph $G$ of order $n$ with vertex set $\{v_1,v_2,\ldots, v_n\}$, let $d_i^+$ and $d_i^-$ denote the out-degree and in-degree of a vertex $v_i$ in $G$, respectively. Let $D^+(G)=diag(d_1^+,d_2^+,\ldots,d_n^+)$ and $D^-(G)=diag(d_1^-,d_2^-,\ldots,d_n^-)$. In this paper we introduce $\widetilde{SL}(G)=\widetilde{D}(G)-S(G)$ to be a new kind of skew Laplacian matrix of $G$, where $\widetilde{D}(G)=D^+(G)-D^-(G)$ and $S(G)$ is the skew-adjacency matrix of $G$, and from which we define the skew Laplacian energy $SLE(G)$ of $G$ as the sum of the norms of all the eigenvalues of $\widet…
A propeller graph is obtained from an $\infty$-graph by attaching a path to the vertex of degree four, where an $\infty$-graph consists of two cycles with precisely one common vertex. In this paper, we prove that all propeller graphs are determined by their Laplacian spectra as well as their signless Laplacian spectra.
Conjecturally, almost all graphs are determined by their spectra. This problem has also been studied for variants such as the spectra of the Laplacian and signless Laplacian. Here we consider the problem of determining graphs with Ihara and Bartholdi zeta functions, which are also computable in polynomial time. These zeta functions are geometrically motivated, but can be viewed as certain generalizations of characteristic polynomials. After discussing some graph properties determined by zeta functions, we show that large classes of cospectral graphs can be distinguished with zeta functions and…
Determining some graph joins by the signless Laplacian spectrum
本文研究由无符号拉普拉斯谱确定的一类图并接结构，给出了$K_1$与若干圈及孤立点并接的图被谱确定的充要条件，并构造了谱同构的例外图，推广了已有结果。
This article focuses on finding the eigenvalues of the Laplacian matrix of the comaximal graph $Γ(\mathbb Z_n)$ of the ring $\mathbb Z_n$ for $n> 2$. We determine the eigenvalues of $Γ(\mathbb Z_n)$ for various $n$ and also provide a procedure to find the eigenvalues of $Γ(\mathbb Z_n)$ for any $n> 2$. We show that $Γ(\mathbb Z_n)$ is Laplacian Integral for $n=p^αq^β$ where $p,q$ are primes and $α, β$ are non-negative integers. The algebraic and vertex connectivity of $Γ(\mathbb Z_n)$ have been shown to be equal for all $n> 2$. An upper bound on the second largest eigenvalue of $Γ(\mathbb Z_n)…
The reciprocal distance Laplacian matrix of a connected graph $G$ is defined as $RD^L(G)=RT(G)-RD(G)$, where $RT(G)$ is the diagonal matrix of reciprocal distance degrees and $RD(G)$ is the Harary matrix. Since $RD^L(G)$ is a real symmetric matrix, we denote its eigenvalues as $λ_1(RD^L(G))\geq λ_2(RD^L(G))\geq \dots \geq λ_n(RD^L(G))$. The largest eigenvalue $λ_1(RD^L(G))$ of $RD^L(G)$ is called the reciprocal distance Laplacian spectral radius. In this article, we prove that the multiplicity of $n$ as a reciprocal distance Laplacian eigenvalue of $RD^L(G)$ is exactly one less than the number…
In this paper, we introduce two new graph operations, namely, the subdivision vertex-edge neighbourhood vertex-corona and the subdivision vertex-edge neighbourhood edge-corona on graphs $G_1$, $G_2$ and $G_3$, and the resulting graphs are denoted by $G_1^S\bowtie (G_2^V\cup G_3^E)$ and $G_1^S\diamondsuit(G_2^V\cup G_3^E)$, respectively. Whereafter, the normalized Laplacian spectra of $G_1^S\bowtie (G_2^V\cup G_3^E)$ and $G_1^S\diamondsuit(G_2^V\cup G_3^E)$ are respectively determined in terms of the corresponding normalized Laplacian spectra of the connected regular graphs $G_{1}$, $G_{2}$ and…
Girth and Laplacian eigenvalue distribution
本文研究了阶数为 $n$、围长为 $g$ 的连通图 $G$ 的 Laplacian 特征值分布，证明了当 $g\ge 4$ 时特定区间内的特征值数量上界为 $n-g$，并刻画了达到该界及相关等式的所有图。
Laplacian Spectrum of the Weakly Zero-Divisor Graph of a Finite Commutative Ring
基于环的局部环分解，将弱零因子图统一表示为完全多部图，计算其拉普拉斯谱的闭式表达式，证明拉普拉斯整性，并给出代数连通度和生成树数的显式公式。
In this paper, we consider continuous-time quantum walks (CTQWs) on finite graphs determined by the Laplacian matrices. By introducing fully interconnected graph decomposition of given graphs, we show a decomposition method for the Laplacian matrices. Using the decomposition method, we show several conditions for graph structure which return probability of CTQW tends to 1 while the number of vertices tends to infinity.
Spectra and Laplacian spectra of arbitrary powers of lexicographic products of graphs
本文研究了任意图的词法幂的谱和拉普拉斯谱，特别探讨了正则图和任意图的特征值特性，并通过实例展示了大规模图的谱分析方法。
A geometric construction of isospectral magnetic graphs
本文利用任意有限磁图作为构建块，通过沿顶点子集逐层收缩副本与框架图，构造出由自然数分拆标记的有限等谱且不同构的磁图族。
Let $A(G)$ and $D(G)$ be the adjacency matrix and the degree diagonal matrix of a graph $G$, respectively. Then $L(G)=D(G)-A(G)$ is called Laplacian matrix of the graph $G$. Let $G$ be a graph with $n$ vertices and $m$ edges. Then the $LI$-matrix of $G$ are defined as $LI(G)=L(G)-\frac{2m}{n}I_n$, where $I_n$ is the identity matrix. In this paper, we are interested in extremal properties of the Ky Fan $k$-norm of the $LI$-matrix of graphs, which is closely related to the well known problems and results in spectral graph theory, such as the Laplacian spectral radius, the Laplacian spread, the s…
Spectra of s-neighbourhood corona of two signed graphs
本研究针对两个符号图的s-邻域冠运算，推导了其邻接谱、拉普拉斯谱和净拉普拉斯谱与原图对应谱的关系，构造了少特征值符号图，并证明该运算结果不被其谱唯一确定。
Signless Laplacian spectral analysis of a class of graph joins
本文研究一类由完全图、圈和匹配边经不交并及连接构成的图，证明其在顶点数足够大时可由无符号拉普拉斯谱唯一确定，并显式计算其谱。
In this paper, it is shown that the graph $T_4(p,q,r)$ is determined by its Laplacian spectrum and there are no two non-isomorphic such graphs which are cospectral with respect to adjacency spectrum.
In 1962, Erdős gave a sufficient condition for Hamilton cycles in terms of the vertex number, edge number, and minimum degree of graphs which generalized Ore's theorem. One year later, Moon and Moser gave an analogous result for Hamilton cycles in balanced bipartite graphs. In this paper we present the spectral analogues of Erdős' theorem and Moon-Moser's theorem, respectively. Let $\mathcal{G}_n^k$ be the class of non-Hamiltonian graphs of order $n$ and minimum degree at least $k$. We determine the maximum (signless Laplacian) spectral radius of graphs in $\mathcal{G}_n^k$ (for large enough $…
Our primary motivation is existence and uniqueness for the obstacle problem on graphs. That is, we look for unique solutions to the problem $Lu = χ_{\{u>0\}}$, where $L$ is the Laplacian matrix associated to a graph, and $u$ is a nonnegative real-valued vector with preassigned zero coordinates and positive coordinates to be determined. In the course of solving this problem, we make a detour into the study of Laplacian matrices themselves. First, we present the row reduced echelon form of such matrices and determine the invertibility of proper square submatrices. Next, we determine eigenvalues …
The nullity of the net Laplacian matrix of a signed graph
本文研究连通符号图的净拉普拉斯矩阵零度，证明了其上下界，并完全确定了零度达到边界时的连通符号图与符号仙人掌图的结构特征。
Contributions in Algebraic Graph Theory
研究代数图论中谱方法在图的谱确定及广义汉明图传递性方面的应用，通过多种方法给出完全二部图等谱特征新证，分类广义汉明图传递参数并给出函数闭式，展示谱方法对理解图结构的作用及后续研究方向。
In a recent paper Gunnells, Scott and Walden have determined the complete spectrum of the Schreier graph on the symmetric group corresponding to the Young subgroup $S_{n-2}\times S_2$ and generated by initial reversals. In particular they find that the first nonzero eigenvalue, or spectral gap, of the Laplacian is always 1, and report that "empirical evidence" suggests that this also holds for the corresponding Cayley graph. We provide a simple proof of this last assertion, based on the decomposition of the Laplacian of Cayley graphs, into a direct sum of irreducible representation matrices of…
A solution to a conjecture on the signless Laplacian spectral radius for $t$-color-critical graphs
本文证明了关于$t$-色临界图$F$的$n$顶点$F$-自由图中符号拉普拉斯谱半径由$K_{t-1}\vee T_{n-t+1,r}$唯一最大化的猜想，采用Zykov对称化类比与对$n$归纳两种新技术。
On the multiplicity of 1 as a Laplacian eigenvalue of a graph
本文研究图的拉普拉斯特征值1的重数，通过约化图方法证明约化树和约化单圈图的重数上界，并完全刻画达到上界的图类。
Discrete Green's functions are the inverses or pseudo-inverses of combinatorial Laplacians. We present compact formulas for discrete Green's functions, in terms of the eigensystems of corresponding Laplacians, for products of regular graphs with or without boundary. Explicit formulas are derived for the cycle, torus, and 3-dimensional torus, as is an inductive formula for the $t$-dimensional torus with $n$ vertices, from which the Green's function can be completely determined in time $O(t n^{2-1/t}\log{n})$. These Green's functions may be used in conjunction with diffusion-like problems on gra…
In this study we investigate the spectra of the family of connected multicone graphs. A multicone graph is defined to be the join of a clique and a regular graph. Let $ r $, $ t $ and $ s $ be natural numbers, and let $ K_r $ denote a complete graph on $ r $ vertices. It is proved that connected multicone graphs $ K_r\bigtriangledown sK_t $, a natural generalization of friendship graphs, are determined by their adjacency spectra as well as their Laplacian spectra. Also, we show that the complement of multicone graphs $ K_r\bigtriangledown sK_t $ are determined by their adjacency spectra, where…
In [DOI:10.1109/ICMEW.2014.6890711], a graph-based denoising is performed by projecting the noisy image to a lower dimensional Krylov subspace of the graph Laplacian, constructed using nonnegative weights determined by distances between image data corresponding to image pixels. We~extend the construction of the graph Laplacian to the case, where some graph weights can be negative. Removing the positivity constraint provides a more accurate inference of a graph model behind the data, and thus can improve quality of filters for graph-based signal processing, e.g., denoising, compared to the stan…
The signless Laplacian spectral radius of graphs without disjoint cliques
研究不含t+1个顶点不交的K_{r+1}拷贝的图的无符号拉普拉斯谱半径极值问题，证明K_t ∨ T_r(n-t)是唯一达到最大谱半径的图。
Which $L$-cospectral graphs have same degree sequences
本文研究了哪些 Laplacian 共谱图具有相同度序列的问题，证明了在特定特征值条件下共谱图必具相同度序列，并据此推导出多扇图及相关复合图由 Laplacian 谱确定的结论。
Periodic discrete graphs with prescribed spectrum
通过构造刷状几何的周期加权图，其离散拉普拉斯算子谱恰好有n个带隙，并证明通过适当选择权重可使带隙端点和谱上界达到指定值。
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