Eleonora Andreotti, Raffaella Mulas
The spectral theory of the normalized Laplacian for chemical hypergraphs is further investigated. The signless normalized Laplacian is introduced and it is shown that its spectrum for classical hypergraphs coincides with the spectrum of the normalized Laplacian for bipartite chemical hypergraphs. Furthermore, the spectra of special families of hypergraphs are established.
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The Signless Laplacian Matrix of Hypergraphs
本文定义超图的无符号拉普拉斯矩阵，利用特征值研究最大度、直径和色数等结构参数，并基于基础超图谱刻画幂超图的完整无符号拉普拉斯谱。
Let $G$ be a simple graph. The signless Laplacian spread of $G$ is defined as the maximum distance of pairs of its signless Laplacian eigenvalues. This paper establishes some new bounds, both lower and upper, for the signless Laplacian spread. Several of these bounds depend on invariant parameters of the graph. We also use a minmax principle to find several lower bounds for this spectral invariant.
Let $G$ be a simple graph. In 1986, Herbert Wilf asked what kind of graphs have an eigenvector with entries formed only by $\pm 1$? In this paper, we answer this question for the adjacency, Laplacian and signless Laplacian matrix of a graph. Besides, we generalize the concept of an exact graph to the adjacency and signless Laplacian matrices. Infinity families of exact graphs for all those matrices are presented.
We generalize the notions of Laplacian and signless Laplacian Estrada index to uniform hypergraphs. For an $r$-uniform hypergraph $H,$ we derive an order $r+1$ trace formula of the (signless) Laplacian tensor of $H.$ Among others by using this trace formula, we obtain lower bounds for the signless Laplacian Estrada index and upper bounds for the Laplacian Estrada index. Moreover, we establish a bound involving both the Laplacian Estrada index and Laplacian energy of a uniform hypergraph.
Let $\mathcal{H}$ be a connected $k$-uniform hypergraph on $n$ vertices and $m$ hyperedges. In [A.~Banerjee, On the spectrum of hypergraph, Linear Algebra and its Application, 614(2021), 82--110], Anirban Banerjee introduced a new adjacency matrix for hypergraphs. In this article we consider the corresponding signless Laplacian matrix $Q(\mathcal{H})$ and discuss about its spectrum.
In this paper we compute bounds for signless Laplacian energy, distance signless Laplacian eigenvalues and signless Laplacian energy of unitary addition Cayley graph G_{n}. We also obtain distance Laplacian eigenvalues and distance Laplacian energy of G_{n}.
We give sharp upper bounds for the ordinary spectral radius and signless Laplacian spectral radius of a uniform hypergraph in terms of the average $2$-degrees or degrees of vertices, respectively, and we also give a lower bound for the ordinary spectral radius. We also compare these bounds with known ones.
We introduce the signless 1-Laplacians and the dual Cheeger constants on simplicial complexes. The connection of its spectrum to the combinatorial properties like independence number, chromatic number and dual Cheeger constant is investigated. Our estimates can be comparable to Hoffman's bounds in virtue of Laplacian on simplicial complexes. An interesting inequality relating multiplicity of the largest eigenvalue, independence number and chromatic number are provided, which could be regarded as a variant version of Lovasz sandwich theorem. Also, the behavior of the operator under the topologi…
In this paper, we study the entries of the principal eigenvector of the signless Laplacian matrix of a hypergraph. More precisely, we obtain bounds for this entries. These bounds are computed trough other important parameters, such as spectral radius, maximum and minimum degree. We also introduce and study a new parameter related to edges of the hypergraph. This parameter is a spectral measure of a structural characteristic that can be thought of as an edge-variant of regularity.
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, we first give the necessary and sufficient conditions for a graph with one main signless Laplacian eigenvalue or two main signless Laplacian eigenvalues, and then characterize the trees and unicyclic graphs with exactly two main signless Laplacian eigenvalues, respectively.
Let $G$ be a connected $m$-uniform hypergraph. In this paper we mainly consider the eigenvectors of the Laplacian or signless Laplacian tensor of $G$ associated with zero eigenvalue, called the first Laplacian or signless Laplacian eigenvectors of $G$. By means of the incidence matrix of $G$, the number of first Laplacian or signless Laplaican (H- or N-)eigenvectors can be get explicitly by solving the Smith normal form of the incidence matrix over $\mathbb{Z}_m$ or $\mathbb{Z}_2$. Consequently, we prove that the number of first Laplacian (H-)eigenvectors is equal to the number of first signle…
The signless Laplacian spectral radius of a graph is the largest eigenvalue of its signless Laplacian. In this paper, we prove that the graph $K_{2}\nabla P_{n-2}$ has the maximal signless Laplacian spectral radius among all planar graphs of order $n\geq 456$.
In this paper, we determine the maximal Laplacian and signless Laplacian spectral radii for graphs with fixed number of vertices and domination number, and characterize the extremal graphs respectively.
We conjecture that the sum of the square roots of the Laplacian eigenvalues of a graph does not exceed the sum of the square roots of its signless Laplacian eigenvalues.
On the Eccentricity Laplacian and Eccentricity Signless Laplacian Matrices of a Graph
本文引入了图的偏心度矩阵的拉普拉斯和无符号拉普拉斯矩阵，探讨了不同图类中偏心度拉普拉斯、偏心度无符号拉普拉斯和偏心度谱之间的等价性，并通过偏心度谱的对称性和这些拉普拉斯矩阵的相似性对E-双图进行谱刻画。
Let $G$ be a simple $r$-regular graph with $n$ vertices and $m$ vertices. We give the signless Laplacian characteristic polynomials of $xyz$-transformations $G^{xyz}$ of $G$ in terms of $n$, $m$, $r$ and the signless Laplacian spectrum of $G$.
In this paper, we obtain the bounds of the extreme eigenvalues of a normalized and signless Laplacian matrices using by their traces. In addition, we determine the bounds for k-th eigenvalues of normalized and signless Laplacian matrices.
In this paper, we study energies associated with hypergraphs. More precisely, we obtain results for the incidence and the singless Laplacian energies of uniform hypergraphs. In particular, we obtain bounds for the incidence energy as functions of well known parameters, such as maximum degree, Zagreb index and spectral radius. We also relate the incidence and signless Laplacian energies of a hypergraph with the adjacency energies of its subdivision graph and line multigraph, respectively. In addition, we compute the signless Laplacian energy for the class of the power hypergraphs.
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.
M. Aouchiche and P. Hansen proposed the distance Laplacian and the distance signless Laplacian of a connected graph [Two Laplacians for the distance matrix of a graph, LAA 439 (2013) 21{33]. In this paper, we obtain three theorems on the sharp upper bounds of the spectral radius of a nonnegative matrix, then apply these theorems to signless Laplacian matrices and the distance signless Laplacian matrices to obtain some sharp bounds on the spectral radius, respectively. We also proposed a known result about the sharp bound of the signless Laplacian spectral radius has a defect.
The signless Laplacian Q and signless edge-Laplacian S of a given graph may or may not be invertible. The Moore-Penrose inverses of Q and S are studied. In particular, using the incidence matrix, we find combinatorial formulas of the Moore- Penrose inverses of Q and S for trees. Also we present combinatorial formulas of the inverses of Q and S for odd unicyclic graphs.
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.
Limit points of (signless) Laplacian spectral radii of linear trees
本研究将Shearer的毛虫树邻接极限点方法适配用于图拉普拉斯谱半径极限点研究，发现原方法存在无极限点的区间，进而推广至线性树拉普拉斯极限点得到更大极限点集，为相关稠密性证明提供工具。
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 att…
On non-bipartite graphs with integral signless Laplacian eigenvalues at most 6
该研究完成了符号拉普拉斯特征值为不超过6的整数的连通非二部图的完整分类工作，属于组合数学领域的图论研究。
The $\mathcal{Q}$-graph of a graph $G$, denoted by $\mathcal{Q}(G)$, is the graph derived from $G$ by plugging a new vertex to each edge of $G$ and adding a new edge between two new vertices which lie on adjacent edges of $G$. In this paper, we consider to study the existence of the signless Laplacian perfect state transfer and signless Laplacian pretty good state transfer in $\mathcal{Q}$-graphs of graphs. We show that, if all the signless Laplacian eigenvalues of a regular graph $G$ are integers, then the $\mathcal{Q}$-graph of $G$ has no signless Laplacian perfect state transfer. We also gi…
The signless Laplacian spectral Turán problems for hypergraphs
本文研究了超图的无符号拉普拉斯谱Turán问题，证明了在特定条件下极值构造可归结为限制问题，并确定了无Fano平面3-均匀超图中谱半径最大的超图为平衡完全二分图。
In this paper, we present a spectral sufficient condition for a graph to be Hamilton-connected in terms of signless Laplacian spectral radius with large minimum degree.
{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 signless Laplacian Estrada index of a graph $G$ is defined as $SLEE(G)=\sum^{n}_{i=1}e^{q_i}$ where $q_1, q_2, \ldots, q_n$ are the eigenvalues of the signless Laplacian matrix of $G$. In this paper, we show that there are exactly two tricyclic graphs with the maximal signless Laplacian Estrada index.
Distribution of signless Laplacian eigenvalues and graph invariants
本文研究了简单图的无符号拉普拉斯特征值分布，揭示了特定区间内的特征值数量与匹配数、直径等图不变量之间的关系。
In this paper, we use a new and correct method to determine the $n$-vertex $k$-trees with the first three largest signless Laplacian indices.
In this paper, we present a sharp upper and lower bounds for the signless Laplacian spectral radius of graphs in terms of clique number. Moreover, the extremal graphs which attain the upper and lower bounds are characterized. In addition, these results disprove the two conjectures on the signless Laplacian spectral radius in [P. Hansen and C. Lucas, Bounds and conjectures for the signless Laplacian index of graphs, Linear Algebra Appl., 432(2010) 3319-3336].
Let G be a simple undirected connected graph. In this paper, new upper bounds on the distance Laplacian spectral radius of G are obtained. Moreover, new lower and upper bounds for the distance signless Laplacian spectral radius of G are derived. Some of the above mentioned bounds are sharp and the graphs attaining the corresponding bound are characterized. Several illustrative examples are included.
The smallest normalized signless $\infty$-Laplacian eigenvalue for non-bipartite connected graphs
本文研究了非二部连通图的最小规范化无符号∞-拉普拉斯特征值，证明其等于加权无符号关联矩阵广义逆的最小∞-范数的倒数，并举例说明。
Let $G$ be a graph of order $n,$ and let $q_{1}(G) \geq ...\geq q_{n}(G) $ be the eigenvalues of the $Q$-matrix of $G$, also known as the signless Laplacian of $G.$ In this paper we give a necessary and sufficient condition for the equality $q_{k}(G) =n-2,$ where $1<k\leq n.$ In particular, this result solves an open problem raised by Wang, Belardo, Huang and Borovicanin. We also show that [ q_{2}(G) \geqδ(G)] and determine that equality holds if and only if $G$ is one of the following graphs: a star, a complete regular multipartite graph, the graph $K_{1,3,3},$ or a complete multipartite grap…
In this paper, we present some spectral sufficient conditions for a graph to be Hamilton-connected in terms of the spectral radius or signless Laplacian spectral radius of the graph. Our results improve some previous work.
A $k$-cyclic graph is a connected graph of order $n$ and size $n+k-1$. In this paper, we determine the maximal signless Laplacian spectral radius and the corresponding extremal graph among all $C_{4}$-free $k$-cyclic graphs of order $n$. Furthermore, we determine the first three unicyclic, and bicyclic, $C_{4}$-free graphs whose spectral radius of the signless Laplacian is maximal. Similar results are obtained for the (combinatorial) Laplacian.
We show several sharp upper and lower bounds for the sum of the largest eigenvalues of the signless Laplacian matrix. These bounds improve and extend previously known bounds.
In this paper, first, we establish a sufficient condition for a bipartite graph to be Hamilton-connected. Furthermore, we also give two sufficient conditions on distance signless Laplacian spectral radius for a graph to be Hamilton-connected and traceable from every vertex, respectively. Last, we obtain a sufficient condition for a graph to be Hamiltonian in terms of the distance signless Laplacian spectral radius of $G^{C}$.
Let G be a simple graph on $n$ vertices and $e(G)$ edges. Consider $Q(G) = D + A$ as the signless Laplacian of $G$, where $A$ is the adjacency matrix and $D$ is the diagonal matrix of the vertices degree of $G$. Let $q_1(G)$ and $q_2(G)$ be the first and the second largest eigenvalues of $Q(G),$ respectively, and denote by $S_{n}^{+}$ the star graph plus one edge. In this paper, we prove that inequality $q_1(G)+ q_2(G) <= e(G)+3$ is tighter for the graph $S_{n}^{+}$ among all firefly graphs and also tighter to $S_{n}^{+}$ than to the graphs $K_{k} \vee \overline{K_{n-k}}$ recently presented by…
We consider the signless $p$-Laplacian of a graph, a generalisation of the usual signless Laplacian (the case $p=2$). We show a Perron-Frobenius property and basic inequalites for the largest eigenvalue and provide upper and lower bounds for the smallest eigenvalue in terms of a parameter related to the bipartiteness. The latter result generalises bounds by Desai and Rao and, interestingly, in the limit $p\to 1$ upper and lower bounds coincide.
In this paper we determine the graph whose least eigenvalue of signless Laplacian attains the minimum or maximum among all connected non-bipartite graphs of fixed order and given number of pendant vertices. Thus we obtain a lower bound and an upper bound for the least eigenvalue of signless Laplacian of a graph in terms of the number of pendent vertices.
Let $G$ be a simple graph with order $n$ and size $m$. The quantity $M_1(G)=\displaystyle\sum_{i=1}^{n}d^2_{v_i}$ is called the first Zagreb index of $G$, where $d_{v_i}$ is the degree of vertex $v_i$, for all $i=1,2,\dots,n$. The signless Laplacian matrix of a graph $G$ is $Q(G)=D(G)+A(G)$, where $A(G)$ and $D(G)$ denote, respectively, the adjacency and the diagonal matrix of the vertex degrees of $G$. Let $q_1\geq q_2\geq \dots\geq q_n\geq 0$ be the signless Laplacian eigenvalues of $G$. The largest signless Laplacian eigenvalue $q_1$ is called the signless Laplacian spectral radius or $Q$-i…
The $α$-Hermitian adjacency matrix $H_α$ of a mixed graph $X$ has been recently introduced. It is a generalization of the adjacency matrix of unoriented graphs. In this paper, we consider a special case of the complex number $α$. This enables us to define an incidence matrix of mixed graphs. Consequently, we define a generalization of line graphs as well as a generalization of the signless Laplacian adjacency matrix of graphs. We then study the spectral properties of the signless Laplacian adjacency matrix of a mixed graph. Lastly, we characterize when the signless Laplacian adjacency matrix o…
The spectral radius (or the signless Laplacian spectral radius) of a general hypergraph is the maximum modulus of the eigenvalues of its adjacency (or its signless Laplacian) tensor. In this paper, we firstly obtain a lower bound of the spectral radius (or the signless Laplacian spectral radius) of general hypergraphs in terms of clique number. Moreover, we present a relation between a homogeneous polynomial and the clique number of general hypergraphs. As an application, we finally obtain an upper bound of the spectral radius of general hypergraphs in terms of clique number.
Let $G$ be an unicyclic graph of order $n$ and let $Q_G(x)= det(xI-Q(G))={matrix} \sum_{i=1}^n (-1)^i φ_i x^{n-i}{matrix}$ be the characteristic polynomial of the signless Laplacian matrix of a graph $G$. We give some transformations of $G$ which decrease all signless Laplacian coefficients in the set $\mathcal{G}(n,m)$. $\mathcal{G}(n,m)$ denotes all n-vertex unicyclic graphs with matching number $m$. We characterize the graphs which minimize all the signless Laplacian coefficients in the set $\mathcal{G}(n,m)$ with odd (resp. even) girth. Moreover, we find the extremal graphs which have mini…
In this article, we generate large families of non-isomorphic and signless Lalacian cospectral graphs using partial transpose on graphs. Our constructions are significantly powerful. More than $70\%$ of non-isomorphic signless-Laplacian cospectral graphs can be generated with partial transpose when number of vertices is $\le 8$. We have also produced numerous examples of non-isomorphic signless Laplacian cospectral graphs.
Distribution of signless Laplacian eigenvalues and degree sequence
本文通过度序列刻画了图的无符号拉普拉斯特征值在特定子区间内的分布，确定了满足某些特征值计数条件的图，并提出了两个开放问题。
In this paper, we investigate (signless) Laplacian spectral characterization of graphs with star components. Also, we prove that the join graph ${\small K_{n-α}-e\vee αK_{1}}$ is $DQS$ for $n-α>3$ and $α\neq3$, and for $α=3$, we show that its only $Q$-cospectral mate is $\bar{K_{1,3}\cup K_{2}\cup (n-6)K_{1}}$.
Let $Z(G)$ be the centre of a finite non-abelian group $G.$ The non-commuting graph of $G$ is a simple undirected graph with vertex set $G\setminus Z(G),$ and two vertices $u$ and $v$ are adjacent if and only if $uv\ne vu.$ In this paper, we investigate the distance, distance (signless) Laplacian spectra of non-commuting graphs of some classes of finite non-abelian groups, and obtain some conditions on a group so that the non-commuting graph is distance, distance (signless) Laplacian integral.
Let $G$ be a simple graph and $Q(G)$ be the signless Laplacian matrix of $G$. Let $S_α(G)$ be the sum of the $α$-th powers of the nonzero eigenvalues of $Q(G)$. We disprove two conjectures by You and Yang on the extremal values of $S_α(G)$ among bipartite graphs and among graphs with bounded connectivity.
Signless Laplacian Estrada index of a graph $G$, defined as $SLEE(G)=\sum^{n}_{i=1}e^{q_i}$, where $q_1, q_2, \cdots, q_n$ are the eigenvalues of the matrix $\mathbf{Q}(G)=\mathbf{D}(G)+\mathbf{A}(G)$. We determine the unique graphs with maximum signless Laplacian Estrada indices among the set of graphs with given number of cut edges, pendent vertices, (vertex) connectivity and edge connectivity.
For a graph $G$, the signless Laplacian Estrada index is defined as $SLEE(G)=\sum^{n}_{i=1}e^{q^{}_i}$,where $q_1, q_2, \dots, q_n$are the eigenvalues of the signless Laplacian matrix of $G$. In this paper, we first characterize the unicyclic graphs with the first two largest and smallest $SLEE$ and then determine the unique unicyclic graph with maximum $SLEE$ among the unicyclic graphs on $n$ vertices with given diameter.
A note on the distance and distance signless Laplacian spectral radius of complements of trees
该研究通过分析广义树移位操作对树补图三类谱半径的增大作用，修正了既有相关定理证明中的歧义。
Let $G$ be a connected non-bipartite graph on $n$ vertices with domination number $γ\le \frac{n+1}{3}$. We investigate the least eigenvalue of the signless Laplacian of $G$, and present a lower bound for such eigenvalue in terms of the domination number $γ$.
Suppose that the vertex set of a connected graph $G$ is $V(G)=\{v_1,\cdots,v_n\}$. Then we denote by $Tr_{G}(v_i)$ the sum of distances between $v_i$ and all other vertices of $G$. Let $Tr(G)$ be the $n\times n$ diagonal matrix with its $(i,i)$-entry equal to $Tr_{G}(v_{i})$ and $D(G)$ be the distance matrix of $G$. Then $Q_{D}(G)=Tr(G)+D(G)$ is the distance signless Laplacian matrix of $G$. The largest eigenvalues of $Q_D(G)$ is called distance signless Laplacian spectral radius of $G$. In this paper we give some graft transformations on distance signless Laplacian spectral radius of the grap…
An interlacing property of the signless Laplacian of threshold graphs
该研究证明了阈值图的无符号拉普拉斯矩阵特征值与顶点度数存在交错性质，并以此为基础验证了无符号Brouwer猜想对阈值图成立，给出了前k大特征值和的上界表达式。
Let $\mathcal{A(}G\mathcal{)},\mathcal{L(}G\mathcal{)}$ and $\mathcal{Q(}% G\mathcal{)}$ be the adjacency tensor, Laplacian tensor and signless Laplacian tensor of uniform hypergraph $G$, respectively. Denote by $λ(\mathcal{T})$ the largest H-eigenvalue of tensor $\mathcal{T}$. Let $H$ be a uniform hypergraph, and $H^{\prime}$ be obtained from $H$ by inserting a new vertex with degree one in each edge. We prove that $λ(\mathcal{Q(}% H^{\prime}\mathcal{)})\leqλ(\mathcal{Q(}H\mathcal{)}).$ Denote by $G^{k}$ the $k$th power hypergraph of an ordinary graph $G$ with maximum degree $Δ\geq2$. We will…
For a simple graph $G$ and a real number $α$ $\left(α\neq 0,1\right) $ the graph invariant $s_α\left(G\right) $ is equal to the sum of powers of signless Laplacian eigenvalues of $G$. In this note, we present some new bounds on $s_α\left(G\right) $. As a result of these bounds, we also give some results on incidence energy.
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