scholarly journals The Laplacian spread of a tree

2008 ◽  
Vol Vol. 10 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Yi-Zheng Fan ◽  
Jing Xu ◽  
Yi Wang ◽  
Dong Liang
Keyword(s):  
Laplacian Matrix ◽  
Largest Eigenvalue ◽  
Smallest Eigenvalue ◽  
International Audience ◽  
The Difference ◽  
The Largest Eigenvalue

Graphs and Algorithms International audience The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the Laplacian matrix of the graph. In this paper, we show that the star is the unique tree with maximal Laplacian spread among all trees of given order, and the path is the unique one with minimal Laplacian spread among all trees of given order.

scholarly journals The Laplacian Spread of Tricyclic Graphs

10.37236/169 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Yanqing Chen ◽  
Ligong Wang
Keyword(s):  
Laplacian Matrix ◽  
Largest Eigenvalue ◽  
Smallest Eigenvalue ◽  
Fixed Order ◽  
Tricyclic Graphs ◽  
The Difference ◽  
The Largest Eigenvalue

The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the Laplacian matrix of the graph. In this paper, we investigate Laplacian spread of graphs, and prove that there exist exactly five types of tricyclic graphs with maximum Laplacian spread among all tricyclic graphs of fixed order.

Ordering cacti with signless Laplacian spread

2018 ◽  
Vol 34 ◽  
pp. 609-619 ◽  
Author(s):  
Zhen Lin ◽  
Shu-Guang Guo
Keyword(s):  
Connected Graph ◽  
Laplacian Matrix ◽  
Largest Eigenvalue ◽  
Smallest Eigenvalue ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian ◽  
The Difference ◽  
The Largest Eigenvalue

A cactus is a connected graph in which any two cycles have at most one vertex in common. The signless Laplacian spread of a graph is defined as the difference between the largest eigenvalue and the smallest eigenvalue of the associated signless Laplacian matrix. In this paper, all cacti of order n with signless Laplacian spread greater than or equal to n − 1/2 are determined.

scholarly journals On the Spectrum of Laplacian Matrix

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Akbar Jahanbani ◽  
Seyed Mahmoud Sheikholeslami ◽  
Rana Khoeilar
Keyword(s):  
Lower Bounds ◽  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Simple Graph ◽  
Upper And Lower Bounds ◽  
Largest Eigenvalue ◽  
Smallest Eigenvalue ◽  
The Matrix ◽  
Vertex Degrees ◽  
The Largest Eigenvalue

Let G be a simple graph of order n . The matrix ℒ G = D G − A G is called the Laplacian matrix of G , where D G and A G denote the diagonal matrix of vertex degrees and the adjacency matrix of G , respectively. Let l 1 G , l n − 1 G be the largest eigenvalue, the second smallest eigenvalue of ℒ G respectively, and λ 1 G be the largest eigenvalue of A G . In this paper, we will present sharp upper and lower bounds for l 1 G and l n − 1 G . Moreover, we investigate the relation between l 1 G and λ 1 G .

scholarly journals Bounds on the $A_{\alpha}$-spread of a graph

2020 ◽  
Vol 36 (36) ◽  
pp. 214-227 ◽  
Author(s):  
Zhen Lin ◽  
Lianying Miao ◽  
Shu-Guang Guo
Keyword(s):  
Real Number ◽  
Adjacency Matrix ◽  
Undirected Graph ◽  
Upper Bounds ◽  
Diagonal Matrix ◽  
Lower And Upper Bounds ◽  
Largest Eigenvalue ◽  
Smallest Eigenvalue ◽  
The Difference ◽  
The Largest Eigenvalue

Let $G$ be a simple undirected graph. For any real number $\alpha \in[0,1]$, Nikiforov defined the $A_{\alpha}$-matrix of $G$ as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the degree diagonal matrix of $G$, respectively. The $A_{\alpha}$-spread of a graph is defined as the difference between the largest eigenvalue and the smallest eigenvalue of the associated $A_{\alpha}$-matrix. In this paper, some lower and upper bounds on $A_{\alpha}$-spread are obtained, which extend the results of $A$-spread and $Q$-spread. Moreover, the trees with the minimum and the maximum $A_{\alpha}$-spread are determined, respectively.

scholarly journals A sharp upper bound on the largest eigenvalue of the Laplacian matrix of a graph

2002 ◽  
Vol 347 (1-3) ◽  
pp. 123-129 ◽  
Author(s):  
Jin-Long Shu ◽  
Yuan Hong ◽  
Kai Wen-Ren
Keyword(s):  
Upper Bound ◽  
Laplacian Matrix ◽  
Largest Eigenvalue ◽  
The Largest Eigenvalue

A note on a conjecture for the distance Laplacian matrix

2016 ◽  
Vol 31 ◽  
pp. 60-68 ◽  
Author(s):  
Celso Marques da Silva ◽  
Maria Aguieiras Alvarez de Freitas ◽  
Renata Raposo Del-Vecchio
Keyword(s):  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Mathematical Journal ◽  
Laplacian Eigenvalue ◽  
Induced Subgraph ◽  
Largest Eigenvalue ◽  
The Largest Eigenvalue

In this note, the graphs of order n having the largest distance Laplacian eigenvalue of multiplicity n −2 are characterized. In particular, it is shown that if the largest eigenvalue of the distance Laplacian matrix of a connected graph G of order n has multiplicity n − 2, then G = S_n or G = K_(p,p), where n = 2p. This resolves a conjecture proposed by M. Aouchiche and P. Hansen in [M. Aouchiche and P. Hansen. A Laplacian for the distance matrix of a graph. Czechoslovak Mathematical Journal, 64(3):751–761, 2014.]. Moreover, it is proved that if G has P_5 as an induced subgraph then the multiplicity of the largest eigenvalue of the distance Laplacian matrix of G is less than n − 3.

ENHANCING THE SYNCHRONIZABILITY OF SCALE-FREE NETWORKS BY ADDING EDGES

2010 ◽  
Vol 21 (01) ◽  
pp. 67-77 ◽  
Author(s):  
SHENG-JUN WANG ◽  
ZHI-XI WU ◽  
HAI-RONG DONG ◽  
GUANRONG CHEN
Keyword(s):  
Nearest Neighbors ◽  
Laplacian Matrix ◽  
New Method ◽  
Scale Free Network ◽  
Largest Eigenvalue ◽  
Scale Free ◽  
Scale Free Networks ◽  
Free Network ◽  
The Largest Eigenvalue

To efficiently enhance the synchronizability of a scale-free network by adding some edges, we numerically study the effect of edge-adding on the spectrum of the network Laplacian matrix. Based on the relation between the largest eigenvalue of the Laplacian matrix and the largest degree of the scale-free network, we show that adding a new edge to the node of largest degree will generally weaken the synchronizability of a scale-free network. We consequently propose a method to effectively enhance the network synchronizability by attaching the new edge to a node whose nearest-neighbors have small degrees. The effect of the new method is analyzed and demonstrated with comparisons.

scholarly journals The Distance Laplacian Spectral Radius of Clique Trees

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Xiaoling Zhang ◽  
Jiajia Zhou
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Minimum Distance ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
Laplacian Spectral Radius ◽  
Largest Eigenvalue ◽  
Clique Trees ◽  
The Largest Eigenvalue

The distance Laplacian matrix of a connected graph G is defined as ℒ G = Tr G − D G , where D G is the distance matrix of G and Tr G is the diagonal matrix of vertex transmissions of G . The largest eigenvalue of ℒ G is called the distance Laplacian spectral radius of G . In this paper, we determine the graphs with maximum and minimum distance Laplacian spectral radius among all clique trees with n vertices and k cliques. Moreover, we obtain n vertices and k cliques.

scholarly journals Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

2021 ◽  
Author(s):  
Jürgen Jost ◽  
Raffaella Mulas ◽  
Florentin Münch
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Spectral Gap ◽  
Complete Bipartite Graph ◽  
Graph Laplacian ◽  
New Method ◽  
Single Edge ◽  
Largest Eigenvalue ◽  
Complement Graph ◽  
The Largest Eigenvalue

AbstractWe offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least $$\frac{n+1}{n-1}$$ n + 1 n - 1 provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $$\frac{n-1}{2}$$ n - 1 2 . With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $$\frac{n-1}{2}$$ n - 1 2 .

scholarly journals On minimum algebraic connectivity of graphs whose complements are bicyclic

2019 ◽  
Vol 17 (1) ◽  
pp. 1490-1502 ◽  
Author(s):  
Jia-Bao Liu ◽  
Muhammad Javaid ◽  
Mohsin Raza ◽  
Naeem Saleem
Keyword(s):  
Connected Graph ◽  
Diffusion Processes ◽  
Laplacian Matrix ◽  
Group Differences ◽  
Algebraic Connectivity ◽  
Connected Graphs ◽  
Bicyclic Graph ◽  
Smallest Eigenvalue ◽  
Unique Graph ◽  
The Stability

Abstract The second smallest eigenvalue of the Laplacian matrix of a graph (network) is called its algebraic connectivity which is used to diagnose Alzheimer’s disease, distinguish the group differences, measure the robustness, construct multiplex model, synchronize the stability, analyze the diffusion processes and find the connectivity of the graphs (networks). A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to its number of vertices plus one. In this paper, firstly the unique graph with a minimum algebraic connectivity is characterized in the class of connected graphs whose complements are bicyclic with exactly three cycles. Then, we find the unique graph of minimum algebraic connectivity in the class of connected graphs $\begin{array}{} {\it\Omega}^c_{n}={\it\Omega}^c_{1,n}\cup{\it\Omega}^c_{2,n}, \end{array}$ where $\begin{array}{} {\it\Omega}^c_{1,n} \end{array}$ and $\begin{array}{} {\it\Omega}^c_{2,n} \end{array}$ are classes of the connected graphs in which the complement of each graph of order n is a bicyclic graph with exactly two and three cycles, respectively.

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