scholarly journals Asymptotic Behaviour of the Number of Eulerian Circuits

10.37236/706 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Mikhail Isaev
Keyword(s):  
Asymptotic Behaviour ◽  
Laplacian Matrix ◽  
Algebraic Connectivity ◽  
Simple Graphs ◽  
Smallest Eigenvalue

We determine the asymptotic behaviour of the number of Eulerian circuits in undirected simple graphs with large algebraic connectivity (the second-smallest eigenvalue of the Laplacian matrix). We also prove some new properties of the Laplacian matrix.

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.

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].

scholarly journals The Least Algebraic Connectivity of Graphs

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Guisheng Jiang ◽  
Guidong Yu ◽  
Jinde Cao
Keyword(s):  
Laplacian Matrix ◽  
Algebraic Connectivity ◽  
Unicyclic Graphs ◽  
Smallest Eigenvalue

The algebraic connectivity of a graph is defined as the second smallest eigenvalue of the Laplacian matrix of the graph, which is a parameter to measure how well a graph is connected. In this paper, we present two unique graphs whose algebraic connectivity attain the minimum among all graphs whose complements are trees, but not stars, and among all graphs whose complements are unicyclic graphs, but not stars adding one edge, respectively.

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.

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.

The Evaluation of Eigenvalues of a Differential Equation Arising in a Problem in Genetics

1962 ◽  
Vol 58 (4) ◽  
pp. 588-593 ◽  
Author(s):  
G. F. Miller ◽  
E. T. Goodwin
Keyword(s):  
Differential Equation ◽  
Asymptotic Behaviour ◽  
Order Differential Equation ◽  
Finite Size ◽  
Second Order ◽  
Random Drift ◽  
Smallest Eigenvalue ◽  
Second Order Differential Equation ◽  
Two Parameters

ABSTRACTThis paper concerns the determination of the smallest eigenvalue of a second order differential equation containing two parameters which arises in problems concerning genic selection under random drift in a population of finite size. A table of values is given, the method of computation is described, and the asymptotic behaviour for large values of one of the parameters is investigated.

scholarly journals Evaluating the Cascade Effect in Interdependent Networks via Algebraic Connectivity

2015 ◽  
Vol 1 (1) ◽  
pp. 55-68 ◽  
Author(s):  
Sotharith Tauch ◽  
William Liu ◽  
Russel Pears
Keyword(s):  
Network Structure ◽  
Topological Structure ◽  
Laplacian Matrix ◽  
Network Robustness ◽  
Node Degree ◽  
Algebraic Connectivity ◽  
Interdependent Networks ◽  
Cascade Effect ◽  
Underlying Network ◽  
Average Node Degree

Understanding how the underlying network structure and interconnectivity impact on the robustness of the interdependent networks is a major challenge in complex networks studies. There are some existing metrics that can be used to measure network robustness. However, different metrics such as the average node degree interprets different characteristic of network topological structure, especially less metrics have been identified to effectively evaluate the cascade performance in interdependent networks. In this paper, we propose to use a combined Laplacian matrix to model the interdependent networks and their interconnectivity, and then use its algebraic connectivity metric as a measure to evaluate its cascading behavior. Moreover, we have conducted extensive comparative studies among different metrics such as the average node degree, and the proposed algebraic connectivity. We have found that the algebraic connectivity metric can describe more accurate and finer characteristics on topological structure of the interdependent networks than other metrics widely adapted by the existing research studies for evaluating the cascading performance in interdependent networks.

scholarly journals Eigenvalue Bounds for the Signless $p$-Laplacian

10.37236/6683 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Elizandro Max Borba ◽  
Uwe Schwerdtfeger
Keyword(s):  
Quadratic Form ◽  
Lower Bounds ◽  
Unit Sphere ◽  
Laplacian Matrix ◽  
Upper And Lower Bounds ◽  
Eigenvalue Bounds ◽  
Smallest Eigenvalue ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian ◽  
Largest Eigenvalues

We consider the signless $p$-Laplacian $Q_p$ of a graph, a generalisation of the quadratic form of the signless Laplacian matrix (the case $p=2$). In analogy to Rayleigh's principle the minimum and maximum of $Q_p$ on the $p$-norm unit sphere are called its smallest and largest eigenvalues, respectively. 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 graph parameter related to the bipartiteness. The latter result generalises bounds by Desai and Rao and, interestingly, at $p=1$ upper and lower bounds coincide.

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