scholarly journals Algebraic Connectivity and Disjoint Vertex Subsets of Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Yan Sun ◽  
Faxu Li
Keyword(s):  
Laplacian Matrix ◽  
Upper Bounds ◽  
Small Eigenvalue ◽  
Algebraic Connectivity

It is well known that the algebraic connectivity of a graph is the second small eigenvalue of its Laplacian matrix. In this paper, we mainly research the relationships between the algebraic connectivity and the disjoint vertex subsets of graphs, which are presented through some upper bounds on algebraic connectivity.

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.

Laplacian Energy of Digraphs and a Minimum Laplacian Energy Algorithm

2015 ◽  
Vol 26 (03) ◽  
pp. 367-380 ◽  
Author(s):  
Xingqin Qi ◽  
Edgar Fuller ◽  
Rong Luo ◽  
Guodong Guo ◽  
Cunquan Zhang
Keyword(s):  
Oriented Graph ◽  
Laplacian Matrix ◽  
Upper Bounds ◽  
Spectral Graph Theory ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
Spectral Graph ◽  
Laplacian Energy ◽  
Energy Formula ◽  
Matrix Formula

In spectral graph theory, the Laplacian energy of undirected graphs has been studied extensively. However, there has been little work yet for digraphs. Recently, Perera and Mizoguchi (2010) introduced the directed Laplacian matrix [Formula: see text] and directed Laplacian energy [Formula: see text] using the second spectral moment of [Formula: see text] for a digraph [Formula: see text] with [Formula: see text] vertices, where [Formula: see text] is the diagonal out-degree matrix, and [Formula: see text] with [Formula: see text] whenever there is an arc [Formula: see text] from the vertex [Formula: see text] to the vertex [Formula: see text] and 0 otherwise. They studied the directed Laplacian energies of two special families of digraphs (simple digraphs and symmetric digraphs). In this paper, we extend the study of Laplacian energy for digraphs which allow both simple and symmetric arcs. We present lower and upper bounds for the Laplacian energy for such digraphs and also characterize the extremal graphs that attain the lower and upper bounds. We also present a polynomial algorithm to find an optimal orientation of a simple undirected graph such that the resulting oriented graph has the minimum Laplacian energy among all orientations. This solves an open problem proposed by Perera and Mizoguchi at 2010.

Resilient Design of Complex Engineered Systems

2013 ◽  
Author(s):  
Hoda Mehrpouyan ◽  
Brandon Haley ◽  
Andy Dong ◽  
Irem Y. Tumer ◽  
Chris Hoyle
Keyword(s):  
Complex Network ◽  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Algebraic Connectivity ◽  
Graph Spectra ◽  
Complex Engineered Systems ◽  
Resilient Design ◽  
Spectra Properties ◽  
Key Driver ◽  
Engineered Systems

This paper presents a complex network and graph spectral approach to calculate the resiliency of complex engineered systems. Resiliency is a key driver in how systems are developed to operate in an unexpected operating environment, and how systems change and respond to the environments in which they operate. This paper deduces resiliency properties of complex engineered systems based on graph spectra calculated from their adjacency matrix representations, which describes the physical connections between components in a complex engineered systems. In conjunction with the adjacency matrix, the degree and Laplacian matrices also have eigenvalue and eigenspectrum properties that can be used to calculate the resiliency of the complex engineered system. One such property of the Laplacian matrix is the algebraic connectivity. The algebraic connectivity is defined as the second smallest eigenvalue of the Laplacian matrix and is proven to be directly related to the resiliency of a complex network. Our motivation in the present work is to calculate the algebraic connectivity and other graph spectra properties to predict the resiliency of the system under design.

scholarly journals Sharp Upper Bounds for the Laplacian Spectral Radius of Graphs

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Houqing Zhou ◽  
Youzhuan Xu
Keyword(s):  
Spectral Radius ◽  
Transient Stability ◽  
Laplacian Matrix ◽  
Schwarz Inequality ◽  
Upper Bounds ◽  
Power Network ◽  
Laplacian Eigenvalue ◽  
Laplacian Spectral Radius ◽  
Wide Range ◽  
Simple Connected Graph

The spectrum of the Laplacian matrix of a network plays a key role in a wide range of dynamical problems associated with the network, from transient stability analysis of power network to distributed control of formations. LetG=(V,E)be a simple connected graph onnvertices and letμ(G)be the largest Laplacian eigenvalue (i.e., the spectral radius) ofG. In this paper, by using the Cauchy-Schwarz inequality, we show that the upper bounds for the Laplacian spectral radius ofG.

TWO SHARP UPPER BOUNDS FOR THE SIGNLESS LAPLACIAN SPECTRAL RADIUS OF GRAPHS

2011 ◽  
Vol 03 (02) ◽  
pp. 185-191 ◽  
Author(s):  
YA-HONG CHEN ◽  
RONG-YING PAN ◽  
XIAO-DONG ZHANG
Keyword(s):  
Spectral Radius ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian

The signless Laplacian matrix of a graph is the sum of its degree diagonal and adjacency matrices. In this paper, we present a sharp upper bound for the spectral radius of the adjacency matrix of a graph. Then this result and other known results are used to obtain two new sharp upper bounds for the signless Laplacian spectral radius. Moreover, the extremal graphs which attain an upper bound are characterized.

Sharp upper bounds on the spectral radius of the signless Laplacian matrix of a graph

2013 ◽  
Vol 219 (10) ◽  
pp. 5025-5032 ◽  
Author(s):  
A. Dilek (Güngör) Maden ◽  
Kinkar Ch. Das ◽  
A. Sinan Çevik
Keyword(s):  
Spectral Radius ◽  
Laplacian Matrix ◽  
Upper Bounds ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian

LAPLACIAN EIGENVALUES OF GRAPHS WITH GIVEN DOMINATION NUMBER

2009 ◽  
Vol 02 (01) ◽  
pp. 71-76 ◽  
Author(s):  
Lihua Feng ◽  
Guihai Yu ◽  
Xiqin Lin
Keyword(s):  
Spectral Radius ◽  
Domination Number ◽  
Upper Bounds ◽  
Algebraic Connectivity ◽  
Laplacian Spectral Radius ◽  
Laplacian Eigenvalues ◽  
Eigenvalues Of Graphs

In this paper, we study the Laplacian eigenvalues of graphs on n vertices with domination number γ and present upper bounds for the Laplacian spectral radius and algebraic connectivity as well, which improve old results apparently.

scholarly journals The Orderings of Bicyclic Graphs and Connected Graphs by Algebraic Connectivity

10.37236/434 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Jianxi Li ◽  
Ji-Ming Guo ◽  
Wai Chee Shiu
Keyword(s):  
Linear Algebra ◽  
Laplacian Matrix ◽  
Algebraic Connectivity ◽  
Connected Graphs ◽  
Unicyclic Graphs ◽  
Smallest Eigenvalue ◽  
Bicyclic Graphs

The algebraic connectivity of a graph $G$ is the second smallest eigenvalue of its Laplacian matrix. Let $\mathscr{B}_n$ be the set of all bicyclic graphs of order $n$. In this paper, we determine the last four bicyclic graphs (according to their smallest algebraic connectivities) among all graphs in $\mathscr{B}_n$ when $n\geq 13$. This result, together with our previous results on trees and unicyclic graphs, can be used to further determine the last sixteen graphs among all connected graphs of order $n$. This extends the results of Shao et al. [The ordering of trees and connected graphs by their algebraic connectivity, Linear Algebra Appl. 428 (2008) 1421-1438].

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