José Guerrero
In this work, we define the Laplacian and Normalized Laplacian energies of vertices in a graph, we derive some of its properties and relate them to combinatorial, spectral and geometric quantities of the graph.
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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.
The Faber-Krahn inequality states that among all open domains with a fixed volume in R^n, the ball minimizes the first Dirichlet eigenvalue of the Laplacian. We study an asymptotic discrete analogue of this for the combinatorial Dirichlet Laplacian acting on induced subgraphs of Z^2. Namely, an induced subgraph G with n vertices is called a minimizing subgraph if it minimizes the first eigenvalue of the combinatorial Dirichlet Laplacian among all induced subgraphs with n vertices. Consider an induced subgraph G and take the interior of the union of closed squares of area 1 about each point of …
We consider quantum state transfer relative to the Laplacian matrix of a graph. Let $N(u)$ denote the set of all neighbors of a vertex $u$ in a graph $G$. A pair of vertices $u$ and $v$ are called twin vertices of $G$ provided $N(u)\setminus\{v \}=N(v)\setminus\{u \}$. We investigate the existence of quantum state transfer between a pair of twin vertices in a graph when the edge between the vertices is perturbed. We find that removal of any set of pairwise non-adjacent edges from a complete graph with a number of vertices divisible by $4$ results Laplacian perfect state transfer (or LPST) at $…
It is known that there is an alternative characterization of characteristic vertices for trees with positive weights on their edges via Perron values and Perron branches. Moreover, the algebraic connectivity of a tree with positive edge weights can be expressed in terms of Perron value. In this article, we consider trees with matrix weights on their edges. More precisely, we are interested in trees with the following classes of matrix edge weights: 1. positive definite matrix weights, 2. lower (or upper) triangular matrix weights with positive diagonal entries. For trees with the above classes…
Let $S_{1}(m, d, k)$ be the $k$-uniform supertree obtained from a loose path $P:v_{1}, e_{1}, v_{2}, \ldots,v_{d}, e_{d}, v_{d+1}$ with length $d$ by attaching $m-d$ edges at vertex $v_{\lfloor\frac{d}{2}\rfloor+1}.$ Let $\mathbb{S}(m,d,k)$ be the set of $k$-uniform supertrees with $m$ edges and diameter $d$ and $q(G)$ be the signless Laplacian spectral radius of a $k$-uniform hypergraph $G$. In this paper, we mainly determine $S_{1}(m,d,k)$ with the largest signless Laplacian spectral radius among all supertrees in $\mathbb{S}(m,d,k)$ for $3\leq d\leq m-1$. Furthermore, we determine the uniqu…
The distance matrix $\mathcal{D}(G)$ of a graph $G$ is the matrix containing the pairwise distances between vertices. The transmission of a vertex $v_i$ in $G$ is the sum of the distances from $v_i$ to all other vertices and $T(G)$ is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, $\mathcal{D}^{\mathcal {L}}(G)=I-T(G)^{-1/2}\mathcal{D}(G) T(G)^{-1/2}$, is introduced. This is analogous to the normalized Laplacian matrix, $\mathcal{L}(G)=I-D(G)^{-1/2}A(G)D(G)^{-1/2}$, where $D(G)$ is the diagonal matrix of degrees of the vertices of the grap…
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 $G$.
Common neighborhood energies and their relations with Zagreb index
本文研究图的公共邻域拉普拉斯能量与第一Zagreb指数的关系，引入CNL-超能量图和CNSL-超能量图概念，证明完全二部图不具备这两种超能量性质，并给出相关能量关系与若干界。
Let G be a graph on n vertices. The Laplacian matrix of G, denoted by L(G), is defined as L(G) = D(G) - A(G), where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the vertex degrees of G. A graph G is said to be L-integral is all eigenvalues of the matrix L(G) are integers. In this paper, we characterize all L-integral non-bipartite graphs among all connected graphs with at most two vertices of degree larger than or equal to three.
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 show that, in the graph spectrum of the normalized graph Laplacian on trees, the eigenvalue 1 and eigenvalues near 1 are strongly related to minimum vertex covers. In particular, for the eigenvalue 1, its multiplicity is related to the size of a minimum vertex cover, and zero entries of its eigenvectors correspond to vertices in minimum vertex covers; while for eigenvalues near 1, their distance to 1 can be estimated from minimum vertex covers; and for the largest eigenvalue smaller than 1, the sign graphs of its eigenvectors take vertices in a minimum vertex cover as representatives.
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.
We characterize all graphs for which there are eigenvectors of the graph Laplacian having all their components in {-1,+1} or {-1,0,+ 1}. Graphs having eigenvectors with components in {-1,+1} are called bivalent and are shown to be the regular bipartite graphs and their extensions obtained by adding edges between vertices with the same value for the given eigenvector. Graphs with eigenvectors with components in {-1,0,+ 1} are called trivalent and are shown to be soft-regular graphs -graphs such that vertices associated with non-zero components have the same degree- and their extensions via some…
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 $G$ be a simple graph with $n$ vertices, $m$ edges having Laplacian eigenvalues $μ_1, μ_2, \dots, μ_{n-1},μ_n=0$. The Laplacian energy $LE(G)$ is defined as $LE(G)=\sum_{i=1}^{n}|μ_i-\overline{d}|$, where $\overline{d}=\frac{2m}{n}$ is the average degree of $G$. Radenković and Gutman conjectured that among all trees of order $n$, the path graph $P_n$ has the smallest Laplacian energy. Let $ \mathcal{T}_{n}(d) $ be the family of trees of order $n$ having diameter $ d $. In this paper, we show that Laplacian energy of any tree $T\in \mathcal{T}_{n}(4)$ is greater than the Laplacian energy of…
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 $p\le 2$.
Graphs can be associated with a matrix according to some rule and we can find the spectrum of a graph with respect to that matrix. Two graphs are cospectral if they have the same spectrum. Constructions of cospectral graphs help us establish patterns about structural information not preserved by the spectrum. We generalize a construction for cospectral graphs previously given for the distance Laplacian matrix to a larger family of graphs. In addition, we show that with appropriate assumptions this generalized construction extends to the adjacency matrix, combinatorial Laplacian matrix, signles…
For any graph, we define a rank-1 operator on a bipartite tensor product space, with components associated to the set of vertices and edges respectively. We show that the partial traces of the operator are the Laplacian and the edge-Laplacian. This provides an interpretation of the von Neumann entropy of the (normalized)\ Laplacian as the amount of quantum entanglement between two systems corresponding to vertices and edges. In this framework, cospectral graphs correspond exactly to local unitarily equivalent pure states. Finally, we introduce the notion of coentropic graphs, that is, graphs w…
Fractional revival between twin vertices
本文刻画了加权图中孪生顶点在三种矩阵下的分数复兴，并应用于双锥体顶点。
Let $G$ be a simple graph. A pendant path of $G$ is a path such that one of its end vertices has degree $1$, the other end has degree $\ge3$, and all the internal vertices have degree $2$. Let $p_k(G)$ be the number of pendant paths of length $k$ of $G$, and $q_k(G)$ be the number of vertices with degree $\ge3$ which are an end vertex of some pendant paths of length $k$. Motivated by the problem of characterizing dendritic trees, N. Saito and E. Woei conjectured that any graph $G$ has some Laplacian eigenvalue with multiplicity at least $p_k(G)-q_k(G)$. We prove a more general result for both …
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.
The Laplacian spread of a graph is the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of the graph. We find that the class of strongly regular graphs attains the maximum of largest eigenvalues, the minimum of second-smallest eigenvalues of Laplacian matrices and hence the maximum of Laplacian spreads among all simple connected graphs of fixed order, minimum degree, maximum degree, minimum size of common neighbors of two adjacent vertices and minimum size of common neighbors of two nonadjacent vertices. Some other extremal graphs are also pr…
Solution to an open problem on Laplacian ratio
本文解决了 Brualdi 和 Goldwasser 提出的公开问题，确定了具有 n 个顶点且直径至少为 k 的树的拉普拉斯比的最小值。
Let $T$ be a tree on $n$ vertices with Laplacian $L_T$ and let $GTS_n$ be the generalized tree shift poset on the set of unlabelled trees on $n$ vertices. Inequalities are known for coefficients of the characteristic polynomial of $L_T$ as we go up the poset $GTS_n$. In this work, we generalize these inequalities to the $q$-Laplacian $L^q_T$ of $T$ and to the coefficients of all immanantal polynomials.
Recently Ayyaswamy [1] have introduced a novel concept of the signless Laplacian Estrada index (after here $SLEE$) associated with a graph $G$. After works, we have identified the unique graph with maximum $SLEE$ with a given parameter such as: number of cut vertices, (vertex) connectivity and edge connectivity. In this paper we continue out characterization for two further parameters; diameter and number of cut vertices.
Let $G$ be a simple graph with $n$ vertices and let $μ_1 \geqslant μ_2 \geqslant...\geqslant μ_{n - 1} \geqslant μ_n = 0$ be the eigenvalues of its Laplacian matrix. The Laplacian Estrada index of a graph $G$ is defined as $LEE (G) = \sum\limits_{i = 1}^n e^{μ_i}$. Using the recent connection between Estrada index of a line graph and Laplacian Estrada index, we prove that the path $P_n$ has minimal, while the star $S_n$ has maximal $LEE$ among trees on $n$ vertices. In addition, we find the unique tree with the second maximal Laplacian Estrada index.
For some positive integer $k$, if the finite cyclic group $\mathbb{Z}_k$ can act freely on a graph $G$, then we say that $G$ is $k$-symmetric. In 1985, Faria showed that the multiplicity of Laplacian eigenvalue 1 is greater than or equal to the difference between the number of pendant vertices and the number of quasi-pendant vertices. But if a graph has a pendant vertex, then it is at most 1-connected. In this paper, we investigate a class of 2-connected $k$-symmetric graphs with a Laplacian eigenvalue 1. We also identify a class of $k$-symmetric graphs in which all Laplacian eigenvalues are i…
For a graph $G$ and a non-zero real number $α$, the graph invariant $S_α(G)$ is the sum of the $α^{th}$ power of the non-zero signless Laplacian eigenvalues of $G$. In this paper, we obtain the sharp bounds of $S_α(G)$ for a connected bipartite graph $G$ on $n$ vertices and a connected graph $G$ on $n$ vertices having a connectivity less than or equal to $k$, respectively, and propose some open problems for future research.
Let $T$ be a tree on $n$ vertices with $q$-Laplacian $L_{T}^{q}$. Let $GTS_n$ be the generalized tree shift poset on the set of unlabelled trees with $n$ vertices. We prove that for all $q \in R$, going up on $GTS_n$ has the following effect: the spectral radius and the second smallest eigenvalue of $L_{T}^{q}$ increase while the smallest eigenvalue of $L_{T}^{q}$ decreases. These generalize known results for eigenvalues of the Laplacian. As a corollary, we obtain consequences about the eigenvalues of $q,t$-Laplacians and exponential distance matrices of trees.
Let $G$ be a simple connected graph on $n$ vertices and $m$ edges. In [Linear Algebra Appl. 435 (2011) 2570-2584], Lima et al. posed the following conjecture on the least eigenvalue $q_n(G)$ of the signless Laplacian of $G$: $\displaystyle q_n(G)\ge {2m}/{(n-1)}-n+2$. In this paper we prove a stronger result: For any graph with $n$ vertices and $m$ edges, we have $\displaystyle q_n(G)\ge {2m}/{(n-2)}-n+1 (n\ge 6)$.
Let $G$ be a graph of order $n$ and $Q_G(x)= det(xI-Q(G))= \sum_{i=1}^n (-1)^i φ_i x^{n-i}$ 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 $\mathcal{B}(n)$ of all $n$-vertex bicyclic graphs. $\mathcal{B}^1(n)$ denotes all n-vertex bicyclic graphs with at least one odd cycle. We show that $B_n^1$ (obtained from $C_4$ by adding one edge between two non-adjacent vertices and adding $n-4$ pendent vertices at the vertex of degree 3) minimizes all the signless Laplacia…
The \emph{distance matrix} of a simple connected graph $G$ is $D(G)=(d_{ij})$, where $d_{ij}$ is the distance between the vertices $i$ and $j$ in $G$. We consider a weighted tree $T$ on $n$ vertices with edge weights are square matrix of same size. The distance $d_{ij}$ between the vertices $i$ and $j$ is the sum of the weight matrices of the edges in the unique path from $i$ to $j$. In this article we establish a characterization for the trees in terms of rank of (matrix) weighted Laplacian matrix associated with it. Then we establish a necessary and sufficient condition for the distance matr…
$p$-Laplacian Operators on Hypergraphs
本文推广了有向图与有向超图上的微分算子，提出面向顶点和（超）弧的梯度、伴随算子及 $p$-拉普拉斯算子的新参数化定义，使其满足连续情形的预期性质。
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…
Laplacian spectrum of weakly zero-divisor graph of the ring $\mathbb{Z}_{n}$
本文研究环 $\mathbb{Z}_{n}$ 的弱零因子图的拉普拉斯谱，证明了该图对任意 $n$ 都是拉普拉斯整的。
Inverse eigenvalue problem for Laplacian matrices of a graph
该研究针对图的广义拉普拉斯矩阵逆特征值问题，探究特定图族与少顶点图的可实现谱、有序重数列表，并从理论与数值层面分析规模归一化加权下该矩阵族的最小方差。
The power graph $\mathcal{G}(G)$ of a group $G$ is a simple graph whose vertices are the elements of $G$ and two distinct vertices are adjacent if one is a power of other. In this paper, we investigate the Laplacian spectrum of the power graph $\mathcal{G}(\mathbb{Z}_{p^m}^n)$ of finite abelian $p$-group $\mathbb{Z}_{p^m}^n$. In particular, we prove that the spectrum of group $\mathbb{Z}_{p^m}^n$ is contained in the Laplacian spectrum of graph $\mathcal{G}(\mathbb{Z}_{p^m}^n)$. For a finite abelian group $G$ whose power graph $\mathcal{G}(G)$ is planar, we also prove that the spectrum of group…
Let $G$ be a finite non abelian group. The centralizer graph of $G$ is a simple undirected graph $Γ_{cent}(G)$, whose vertices are the proper centralizers of $G$ and two vertices are adjacent if and only if their cardinalities are identical {\rm\cite{omer}}. The complement of the centralizer graph is called the co-centralizer graph. In this paper, we investigate the adjacency and (signless) Laplacian spectra of centralizer and co-centralizer graphs of some classes of finite non-abelian groups and obtain some conditions on a group so that the centralizer and co-centralizer graphs are adjacency,…
Let $F_k$ denote the $k$-fan consisting of $k$ triangles which intersect in exactly one common vertex, and $S_{n,k}$ the complete split graph of order $n$ consisting of a clique on $k$ vertices and an independent set on the remaining vertices in which each vertex of the clique is adjacent to each vertex of the independent set. In this paper, it is shown that $S_{n,k}$ is the unique graph attaining the maximum signless Laplacian spectral radius among all graphs of order $n$ containing no $F_k$, provided that $k\geq 2$ and $n\geq 3k^2-k-2$.
We introduce a new operation on a class of graphs with the property that the Laplacian eigenvalues of the input and output graphs are related. Based on this operation, we obtain a family of order (square root of n) noncospectral unicyclic graphs on n vertices with the same Laplacian energy.
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…
Let $G$ be a graph with $n$ vertices. We denote the largest signless Laplacian eigenvalue of $G$ by $q_1(G)$ and Laplacian eigenvalues of $G$ by $μ_1(G)\ge\cdots\geμ_{n-1}(G)\geμ_n(G)=0$. It is a conjecture on Laplacian spread of graphs that $μ_1(G)-μ_{n-1}(G)\le n-1$ or equivalently $μ_1(G)+μ_1(\Gb)\le2n-1$. We prove the conjecture for bipartite graphs. Also we show that for any bipartite graph $G$, $μ_1(G)μ_1(\Gb)\le n(n-1)$. Aouchiche and Hansen [A survey of Nordhaus--Gaddum type relations, Discrete Appl. Math. 161 (2013), 466--546] conjectured that %for any graph $G$ with $n$ vertices, $q_…
We develop the theory of fractional revival in the quantum walk on a graph using its Laplacian matrix as the Hamiltonian. We first give a spectral characterization of Laplacian fractional revival, which leads to a polynomial time algorithm to check this phenomenon and find the earliest time when it occurs. We then apply the characterization theorem to special families of graphs. In particular, we show that no tree admits Laplacian fractional revival except for the paths on two and three vertices, and the only graphs on a prime number of vertices that admit Laplacian fractional revival are doub…
Laplacian state transfer in graphs with involutions
本文研究了具有可能环的图中态传输，通过拉普拉斯矩阵作为哈密顿量，证明了环的添加对态传输的影响，并展示了在某些图中态传输的条件。
The distance matrix $\mathcal{D}$ of a connected graph $G$ is the matrix indexed by the vertices of $G$ which entry $\mathcal{D}_{i,j}$ equals the distance between the vertices $v_i$ and $v_j$. The distance signless Laplacian matrix $\mathcal{Q}(G)$ of graph $G$ is defined as $\mathcal{Q}(G)=Diag(Tr)+\mathcal{D}(G)$, where $Diag(Tr)$ is the diagonal matrix of the vertex transmissions in $G$. The largest eigenvalue of $\mathcal{Q}(G)$ is called the distance signless Laplacian spectral radius of $G$, written as $η_1(G)$. And a perfect matching in a graph is a set of disadjacent edges covering ev…
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.
This paper further investigates \emph{Laplacian simplices}. A construction by Braun and the first author associates to a simple connected graph $G$ a simplex $\cP_G$ whose vertices are the rows of the Laplacian matrix of $G$. In this paper we associate to a reflexive $\cP_G$ a duality-preserving linear code $\cC(\cP_G)$. This new perspective allows us to build upon previous results relating graphical properties of $G$ to properties of the polytope $\cP_G$. In particular, we make progress towards a graphical characterization of reflexive $\cP_G$ using techniques from Ehrhart theory. We provide …
We study the Laplacian spectrum of token graphs, also called symmetric powers of graphs. The $k$-token graph $F_k(G)$ of a graph $G$ is the graph whose vertices are the $k$-subsets of vertices from $G$, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in $G$. In this paper, we give a relationship between the Laplacian spectra of any two token graphs of a given graph. In particular, we show that, for any integers $h$ and $k$ such that $1\le h\le k\le \frac{n}{2}$, the Laplacian spectrum of $F_h(G)$ is contained in the Laplacian spectrum of $F_k(G)$.…
Let $G_1$ and $G_2$ be two graphs on disjoint sets of $n_1$ and $n_2$ vertices, respectively. The corona of graphs $G_1$ and $G_2$, denoted by $G_1\circ G_2$, is the graph formed from one copy of $G_1$ and $n_1$ copies of $G_2$ where the $i$-th vertex of $G_1$ is adjacent to every vertex in the $i$-th copy of $G_2$. The neighborhood corona of $G_1$ and $G_2$, denoted by $G_1\diamond G_2$, is the graph obtained by taking one copy of $G_1$ and $n_1$ copies of $G_2$ and joining every neighbor of the $i$-th vertex of $G_1$ to every vertex in the $i$-th copy of $G_2$ by a new edge. In this paper, t…
Faber-Krahn inequalities for first Dirichlet eigenvalues of combinatorial $p$-Laplacian on graphs with boundary
本文研究了图上组合p-拉普拉斯算子的第一狄利克雷特征值的费伯-克兰不等式，证明当p>1时，具有n个顶点或n条边的连通图的最小第一p-狄利克雷特征值仅在 tadpole 图T_{n,3}上取得。
Given a connected graph $G$, the toughness $τ_G$ is defined as the minimum value of the ratio $|S|/ω_{G-S}$, where $S$ ranges over all vertex cut sets of $G$, and $ω_{G-S}$ is the number of connected components in the subgraph $G-S$ obtained by deleting all vertices of $S$ from $G$. In this paper, we provide a lower bound for the toughness $τ_G$ in terms of the maximum degree, minimum degree and normalized Laplacian eigenvalues of $G$. This can be viewed as a slight generalization of Brouwer's toughness conjecture, which was confirmed by Gu (2021). Furthermore, we give a characterization of th…
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。
Some bounds on the Laplacian eigenvalues of token graphs
本文研究令牌图的拉普拉斯特征值，通过较小规模的 h-令牌图给出 k-令牌图特征值的下界，并证明当原图代数连通度足够大时，所有不超过 k 的令牌图代数连通度均与原图相同。
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.
Let $f(D(i, j), d_i, d_j)$ be a real function symmetric in $i$ and $j$ with the property that $f(d, (1+o(1))np, (1+o(1))np)=(1+o(1))f(d, np, np)$ for $d=1,2$. Let $G$ be a graph, $d_i$ denote the degree of a vertex $i$ of $G$ and $D(i, j)$ denote the distance between vertices $i$ and $j$ in $G$. In this paper, we define the $f$-weighted Laplacian matrix for random graphs in the Erd$\ddot{o}$s-R$\acute{e}$nyi random graph model $\mathcal{G}_{n, p}$, where $p\in (0, 1)$ is fixed. Four weighted Laplacian type energies: the weighted Laplacian energy $\mathscr{LE}_f(G)$, weighted signless Laplacian…
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 $Q_{D}(G)=Tr(G)+D(G)$ and $L_{D}(G)=Tr(G)-D(G)$ are respectively the distance signless Laplacian matrix and distance Laplacian matrix of $G$. The largest eigenvalues $ρ_{Q}(G)$ and $ρ_{L}(G)$ of $Q_D(G)$ and $L_D(G)$ are respectively called distance signless Laplacian spectral…
Let $G=(V(G) ,E(G))$ be a digraph without loops and multiarcs, where $V(G)=\{v_1,v_2,\ldots,v_n\}$ and $E(G)$ are the vertex set and the arc set of $G$, respectively. Let $d_i^{+}$ be the outdegree of the vertex $v_i$. Let $A(G)$ be the adjacency matrix of $G$ and $D(G)=\textrm{diag}(d_1^{+},d_2^{+},\ldots,d_n^{+})$ be the diagonal matrix with outdegrees of the vertices of $G$. Then we call $Q(G)=D(G)+A(G)$ the signless Laplacian matrix of $G$. The spectral radius of $Q(G)$ is called the signless Laplacian spectral radius of $G$, denoted by $q(G)$. In this paper, some upper bounds for $q(G)$ a…
The power graph $P(Ω)$ of a group $Ω$ is a graph with the vertex set $Ω$ such that two distinct vertices form an edge if and only if one of them is an integral power of the other. In this article, we determine the power graph of the group $\mathcal{G} = \langle s,r \, : r^{2^kp} = s^2 = e,~ srs^{-1} = r^{2^{k-1}p-1}\rangle$. Further, we compute its characteristic polynomial for the adjacency, Laplacian, and signless Laplacian matrices associated with this power graph. In addition, we determine its spectrum, Laplacian spectrum, and Laplacian energy.
We conjecture a new lower bound on the algebraic connectivity of a graph that involves the number of vertices of high eccentricity in a graph. We prove that this lower bound implies a strengthening of the Laplacian Spread Conjecture. We discuss further conjectures, also strengthening the Laplacian Spread Conjecture, that include a conjecture for simple graphs and a conjecture for weighted graphs.
Quantum state transfer between twins in weighted graphs
本文研究加权图中孪生顶点间的量子态传输，刻画了周期性、完美态传输和相当好的态传输的条件，并应用于正则图上的双锥图。
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