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 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.

Research of Edge Centrality Based on the Algebraic Connectivity

2014 ◽  
Vol 543-547 ◽  
pp. 3636-3640
Author(s):  
En Jun Xing ◽  
Fu Qiang Zhao ◽  
Shuo Zhang ◽  
Xin Yu Ge
Keyword(s):  
Complex Networks ◽  
Information Transmission ◽  
Empirical Analysis ◽  
Connected Graph ◽  
Time Complexity ◽  
Laplacian Matrix ◽  
Medium Size ◽  
Algebraic Connectivity ◽  
Real Network ◽  
Second Eigenvalue

In connected graph, edge centrality represents the importance of edge and loading degree in the process of information transmission. The second eigenvalue of Laplacian matrix decides connectivity of complex networks. We propose edge centrality model and cut model based on minimization model of the algebraic connectivity. Edge centrality function is derived in order to calculate edge centrality. Cut model deletes k edges at an iteration whose algebraic connectivity of complex networks decreased fastest. Choice of k is based on edge sparse degree of complex networks. By empirical analysis of real network, the implications of edge with higher centrality and its vertex coincide with the fact. The model can be used in medium-size networks with lower time complexity and higher efficiency.

Minimal extremal graphs for addition of algebraic connectivity and independence number of connected graphs

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5545-5551
Author(s):  
Kinkar Das ◽  
Muhuo Liu
Keyword(s):  
Lower Bound ◽  
Independence Number ◽  
Laplacian Matrix ◽  
Algebraic Connectivity ◽  
Extremal Graphs ◽  
Laplacian Eigenvalue ◽  
Connected Graphs ◽  
Minimum Value ◽  
Vertex Degrees ◽  
Simple Connected Graph

Let G = (V,E) be a simple connected graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the Laplacian matrix of graph G is L(G) = D(G)-A(G). Let a(G) and ?(G), respectively, be the second smallest Laplacian eigenvalue and the independence number of graph G. In this paper, we characterize the extremal graph with second minimum value for addition of algebraic connectivity and independence number among all connected graphs with n ? 6 vertices (Actually, we can determine the p-th minimum value of a(G)+ ?(G) under certain condition when p is small). Moreover, we present a lower bound to the addition of algebraic connectivity and radius of connected graphs.

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.

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 The normalized distance Laplacian

2021 ◽  
Vol 9 (1) ◽  
pp. 1-18
Author(s):  
Carolyn Reinhart
Keyword(s):  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(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 graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

scholarly journals All adapted topologies are equal

2020 ◽  
Vol 178 (3-4) ◽  
pp. 1125-1172
Author(s):  
Julio Backhoff-Veraguas ◽  
Daniel Bartl ◽  
Mathias Beiglböck ◽  
Manu Eder
Keyword(s):  
Stochastic Processes ◽  
Weak Topology ◽  
Equilibrium Problems ◽  
Diffusion Processes ◽  
Temporal Structure ◽  
Wasserstein Distance ◽  
Optimal Stopping Problems ◽  
Reward Functions ◽  
Nested Distance ◽  
The Stability

Abstract A number of researchers have introduced topological structures on the set of laws of stochastic processes. A unifying goal of these authors is to strengthen the usual weak topology in order to adequately capture the temporal structure of stochastic processes. Aldous defines an extended weak topology based on the weak convergence of prediction processes. In the economic literature, Hellwig introduced the information topology to study the stability of equilibrium problems. Bion–Nadal and Talay introduce a version of the Wasserstein distance between the laws of diffusion processes. Pflug and Pichler consider the nested distance (and the weak nested topology) to obtain continuity of stochastic multistage programming problems. These distances can be seen as a symmetrization of Lassalle’s causal transport problem, but there are also further natural ways to derive a topology from causal transport. Our main result is that all of these seemingly independent approaches define the same topology in finite discrete time. Moreover we show that this ‘weak adapted topology’ is characterized as the coarsest topology that guarantees continuity of optimal stopping problems for continuous bounded reward functions.

scholarly journals The stability index of a cactus

1977 ◽  
Vol 24 (2) ◽  
pp. 170-183
Author(s):  
Douglas D. Grant ◽  
D. A. Holton
Keyword(s):  
Connected Graph ◽  
Stability Index ◽  
The Stability

AbstractWe show that ifGis a connected graph of ordernsuch that no line lies in more than one cycle (in other words,Gis a cactus of ordern), then the stability index ofGis one of the integers 0, 1,n−7,n−6,n−5,n−4 orn.

2-connected graphs with the minimum algebraic connectivity

2021 ◽  
pp. 1-6
Author(s):  
Guanglong Yu ◽  
Lin Sun ◽  
Hailiang Zhang ◽  
Yarong Wu
Keyword(s):  
Algebraic Connectivity ◽  
Connected Graphs
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