scholarly journals Partitions of networks that are robust to vertex permutation dynamics

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Gary Froyland ◽  
Eric Kwok
Keyword(s):  
Spectral Method ◽  
Spectral Methods ◽  
Laplacian Matrix ◽  
Minimum Cut ◽  
General Sequence ◽  
Connected Graphs ◽  
Connectivity Structure ◽  
Fundamental Information ◽  
Minimum Cut Problem

AbstractMinimum disconnecting cuts of connected graphs provide fundamental information about the connectivity structure of the graph. Spectral methods are well-known as stable and efficient means of finding good solutions to the balanced minimum cut problem. In this paper we generalise the standard balanced bisection problem for static graphs to a new “dynamic balanced bisection problem”, in which the bisecting cut should be minimal when the vertex-labelled graph is subjected to a general sequence of vertex permutations. We extend the standard spectral method for partitioning static graphs, based on eigenvectors of the Laplacian matrix of the graph, by constructing a new dynamic Laplacian matrix, with eigenvectors that generate good solutions to the dynamic minimum cut problem. We formulate and prove a dynamic Cheeger inequality for graphs, and demonstrate the effectiveness of the dynamic Laplacian matrix for both structured and unstructured graphs.

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 A Linear Time Algorithm for a Variant of the MAX CUT Problem in Series Parallel Graphs

2017 ◽  
Vol 2017 ◽  
pp. 1-4 ◽  
Author(s):  
Brahim Chaourar
Keyword(s):  
Linear Time ◽  
Maximum Flow ◽  
Time Algorithm ◽  
Minimum Cut ◽  
Linear Time Algorithm ◽  
Induced Subgraphs ◽  
Positive Weight ◽  
Max Cut Problem ◽  
In Series ◽  
Minimum Cut Problem

Given a graph G=V,E, a connected sides cut U,V\U or δU is the set of edges of E linking all vertices of U to all vertices of V\U such that the induced subgraphs GU and GV\U are connected. Given a positive weight function w defined on E, the maximum connected sides cut problem (MAX CS CUT) is to find a connected sides cut Ω such that wΩ is maximum. MAX CS CUT is NP-hard. In this paper, we give a linear time algorithm to solve MAX CS CUT for series parallel graphs. We deduce a linear time algorithm for the minimum cut problem in the same class of graphs without computing the maximum flow.

A fast algorithm for the generalized parametric minimum cut problem and applications

Algorithmica ◽  
1992 ◽  
Vol 7 (1-6) ◽  
pp. 499-519 ◽  
Author(s):  
Dan Gusfield ◽  
Charles Martel
Keyword(s):  
Fast Algorithm ◽  
Minimum Cut ◽  
Minimum Cut Problem

scholarly journals Laplacian integral graphs with a given degree sequence constraint

2021 ◽  
Vol 40 (6) ◽  
pp. 1431-1448
Author(s):  
Ansderson Fernandes Novanta ◽  
Carla Silva Oliveira ◽  
Leonardo de Lima
Keyword(s):  
Adjacency Matrix ◽  
Degree Sequence ◽  
Laplacian Matrix ◽  
Bipartite Graphs ◽  
Diagonal Matrix ◽  
Connected Graphs ◽  
Sequence Constraint ◽  
The Matrix ◽  
Vertex Degrees ◽  
Integral Graphs

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 if all eigenvalues of the matrix L(G) are integers. In this paper, we characterize all Lintegral non-bipartite graphs among all connected graphs with at most two vertices of degree larger than or equal to three.

Max Horn SAT and the minimum cut problem in directed hypergraphs

1998 ◽  
Vol 80 (2) ◽  
pp. 213-237 ◽  
Author(s):  
G. Gallo ◽  
C. Gentile ◽  
D. Pretolani ◽  
G. Rago
Keyword(s):  
Minimum Cut ◽  
Directed Hypergraphs ◽  
Minimum Cut Problem

scholarly journals Optimal Estimation of the Climatological Mean

2009 ◽  
Vol 22 (18) ◽  
pp. 4845-4859 ◽  
Author(s):  
Balachandrudu Narapusetty ◽  
Timothy DelSole ◽  
Michael K. Tippett
Keyword(s):  
Spectral Method ◽  
Spectral Methods ◽  
Cross Validation ◽  
Optimal Estimation ◽  
Sea Surface ◽  
Least Squares Fit ◽  
Straightforward Consequence ◽  
Independent Parameters ◽  
Special Case ◽  
Validation Experiments

Abstract This paper shows theoretically and with examples that climatological means derived from spectral methods predict independent data with less error than climatological means derived from simple averaging. Herein, “spectral methods” indicates a least squares fit to a sum of a small number of sines and cosines that are periodic on annual or diurnal periods, and “simple averaging” refers to mean averages computed while holding the phase of the annual or diurnal cycle constant. The fact that spectral methods are superior to simple averaging can be understood as a straightforward consequence of overfitting, provided that one recognizes that simple averaging is a special case of the spectral method. To illustrate these results, the two methods are compared in the context of estimating the climatological mean of sea surface temperature (SST). Cross-validation experiments indicate that about four harmonics of the annual cycle are adequate, which requires estimation of nine independent parameters. In contrast, simple averaging of daily SST requires estimation of 366 parameters—one for each day of the year, which is a factor of 40 more parameters. Consistent with the greater number of parameters, simple averaging poorly predicts samples that were not included in the estimation of the climatological mean, compared to the spectral method. In addition to being more accurate, the spectral method also accommodates leap years and missing data simply, results in a greater degree of data compression, and automatically produces smooth time series.

scholarly journals Network flow optimization for restoration of images

2002 ◽  
Vol 2 (4) ◽  
pp. 199-218 ◽  
Author(s):  
Boris A. Zalesky
Keyword(s):  
Ising Model ◽  
Network Flow ◽  
Minimum Cut ◽  
Optimization Approach ◽  
Gray Scale ◽  
Flow Optimization ◽  
Binary Images ◽  
Statistical Background ◽  
Network Flow Optimization ◽  
Minimum Cut Problem

The network flow optimization approach is offered for restoration of gray-scale and color images corrupted by noise. The Ising models are used as a statistical background of the proposed method. We present the new multiresolution network flow minimum cut algorithm, which is especially efficient in identification of the maximum a posteriori (MAP) estimates of corrupted images. The algorithm is able to compute the MAP estimates of large-size images and can be used in a concurrent mode. We also consider the problem of integer minimization of two functions,U1(x)=λ∑i|yi−xi|+∑i,j  βi,j|xi−xj|andU2(x)=∑i λi (yi−xi)2+∑i,j  βi,j (xi−xj)2, with parametersλ,λi,βi,j>0and vectorsx=(x1,…,xn),y=(y1,…,yn)∈{0,…,L−1}n. Those functions constitute the energy ones for the Ising model of color and gray-scale images. In the caseL=2, they coincide, determining the energy function of the Ising model of binary images, and their minimization becomes equivalent to the network flow minimum cut problem. The efficient integer minimization ofU1(x),U2(x)by the network flow algorithms is described.

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