scholarly journals Estrada Index and Laplacian Estrada Index of Random Interdependent Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1063
Author(s):  
Yilun Shang
Keyword(s):  
Spectral Property ◽  
Random Graph ◽  
Lower Bounds ◽  
Graph Model ◽  
Upper Bounds ◽  
Simple Graph ◽  
Stochastic Block Model ◽  
Laplacian Matrices ◽  
Estrada Index ◽  
Multipartite Graph

Let G be a simple graph of order n. The Estrada index and Laplacian Estrada index of G are defined by E E ( G ) = ∑ i = 1 n e λ i ( A ( G ) ) and L E E ( G ) = ∑ i = 1 n e λ i ( L ( G ) ) , where { λ i ( A ( G ) ) } i = 1 n and { λ i ( L ( G ) ) } i = 1 n are the eigenvalues of its adjacency and Laplacian matrices, respectively. In this paper, we establish almost sure upper bounds and lower bounds for random interdependent graph model, which is fairly general encompassing Erdös-Rényi random graph, random multipartite graph, and even stochastic block model. Our results unravel the non-triviality of interdependent edges between different constituting subgraphs in spectral property of interdependent graphs.

scholarly journals A Note on the Warmth of Random Graphs with Given Expected Degrees

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Statistical Mechanics ◽  
Chromatic Number ◽  
Random Graph ◽  
Lower Bounds ◽  
Random Graphs ◽  
Upper Bound ◽  
Degree Sequence ◽  
Graph Model ◽  
Upper And Lower Bounds ◽  
Random Graph Model

We consider the random graph modelG(w)for a given expected degree sequencew=(w1,w2,…,wn). Warmth, introduced by Brightwell and Winkler in the context of combinatorial statistical mechanics, is a graph parameter related to lower bounds of chromatic number. We present new upper and lower bounds on warmth ofG(w). In particular, the minimum expected degree turns out to be an upper bound of warmth when it tends to infinity and the maximum expected degreem=O(nα)with0<α<1/2.

scholarly journals Improve Some of Bounds for the Randic Estrada Index of Graphs

2020 ◽  
Author(s):  
Akbar Jahanbani
Keyword(s):  
Lower Bounds ◽  
Upper Bounds ◽  
Randić Index ◽  
Lower And Upper Bounds ◽  
Graph Invariants ◽  
Randic Index ◽  
Estrada Index ◽  
Eigenvalues Of Graphs

Let G be a graph with n vertices and let 1; 2; : : : ; n be the eigenvalues of Randic matrix. The Randic Estrada index of G is REE(G) = &Oacute;n i=1 ei . In this paper, we establish lower and upper bounds for Randic index in terms of graph invariants such as the number of vertices and eigenvalues of graphs and improve some previously published lower bounds.

scholarly journals Some Properties on Estrada Index of Folded Hypercubes Networks

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Jia-Bao Liu ◽  
Xiang-Feng Pan ◽  
Jinde Cao
Keyword(s):  
Adjacency Matrix ◽  
Upper Bounds ◽  
Simple Graph ◽  
Spectral Graph Theory ◽  
Lower And Upper Bounds ◽  
Estrada Index ◽  
Spectral Graph ◽  
Explicit Formulae ◽  
Folded Hypercube ◽  
Hypercube Networks

LetGbe a simple graph withnvertices and letλ1,λ2,…,λnbe the eigenvalues of its adjacency matrix; the Estrada indexEEGof the graphGis defined as the sum of the termseλi,  i=1,2,…,n. Then-dimensional folded hypercube networksFQnare an important and attractive variant of then-dimensional hypercube networksQn, which are obtained fromQnby adding an edge between any pair of vertices complementary edges. In this paper, we establish the explicit formulae for calculating the Estrada index of the folded hypercubes networksFQnby deducing the characteristic polynomial of the adjacency matrix in spectral graph theory. Moreover, some lower and upper bounds for the Estrada index of the folded hypercubes networksFQnare proposed.

scholarly journals On the Number of Non-Zero Elements of Joint Degree Vectors

10.37236/6385 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Éva Czabarka ◽  
Johannes Rauh ◽  
Kayvan Sadeghi ◽  
Taylor Short ◽  
László Székely
Keyword(s):  
Random Graph ◽  
Upper Bound ◽  
Graph Model ◽  
Upper Bounds ◽  
Exponential Random Graph Model ◽  
Lower And Upper Bounds ◽  
Sufficient Statistics ◽  
Random Graph Model ◽  
Exponential Random Graph ◽  
Vertex Graph

Joint degree vectors give the number of edges between vertices of degree $i$ and degree $j$ for $1\le i\le j\le n-1$ in an $n$-vertex graph. We find lower and upper bounds for the maximum number of nonzero elements in a joint degree vector as a function of $n$. This provides an upper bound on the number of estimable parameters in the exponential random graph model with bidegree-distribution as its sufficient statistics.

scholarly journals A Note on Some Bounds of the α-Estrada Index of Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Yang Yang ◽  
Lizhu Sun ◽  
Changjiang Bu
Keyword(s):  
Lower Bounds ◽  
Adjacency Matrix ◽  
Simple Graph ◽  
Upper And Lower Bounds ◽  
Estrada Index ◽  
Degree Matrix

Let G be a simple graph with n vertices. Let A~αG=αDG+1−αAG, where 0≤α≤1 and AG and DG denote the adjacency matrix and degree matrix of G, respectively. EEαG=∑i=1neλi is called the α-Estrada index of G, where λ1,⋯,λn denote the eigenvalues of A~αG. In this paper, the upper and lower bounds for EEαG are given. Moreover, some relations between the α-Estrada index and α-energy are established.

Random graphs

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Large Scale ◽  
Random Network ◽  
Graph Model ◽  
Clustering Coefficient ◽  
Giant Component ◽  
Random Graph Model ◽  
Definition Of ◽  
Average Size

An introduction to the mathematics of the Poisson random graph, the simplest model of a random network. The chapter starts with a definition of the model, followed by derivations of basic properties like the mean degree, degree distribution, and clustering coefficient. This is followed with a detailed derivation of the large-scale structural properties of random graphs, including the position of the phase transition at which a giant component appears, the size of the giant component, the average size of the small components, and the expected diameter of the network. The chapter ends with a discussion of some of the shortcomings of the random graph model.

scholarly journals A Note on the Estrada Index of the Aα-Matrix

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 811
Author(s):  
Jonnathan Rodríguez ◽  
Hans Nina
Keyword(s):  
Lower Bounds ◽  
Adjacency Matrix ◽  
Estrada Index

Let G be a graph on n vertices. The Estrada index of G is an invariant that is calculated from the eigenvalues of the adjacency matrix of a graph. V. Nikiforov studied hybrids of A(G) and D(G) and defined the Aα-matrix for every real α∈[0,1] as: Aα(G)=αD(G)+(1−α)A(G). In this paper, using a different demonstration technique, we present a way to compare the Estrada index of the Aα-matrix with the Estrada index of the adjacency matrix of the graph G. Furthermore, lower bounds for the Estrada index are established.

scholarly journals Multilevel Monte Carlo methods and lower–upper bounds in initial margin computations

2020 ◽  
Vol 26 (2) ◽  
pp. 131-161
Author(s):  
Florian Bourgey ◽  
Stefano De Marco ◽  
Emmanuel Gobet ◽  
Alexandre Zhou
Keyword(s):  
Monte Carlo ◽  
Lower Bounds ◽  
Asymptotic Optimality ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Independent Random Variables ◽  
Outer Function ◽  
Control Problems ◽  
Multilevel Monte Carlo ◽  
Primal Dual

AbstractThe multilevel Monte Carlo (MLMC) method developed by M. B. Giles [Multilevel Monte Carlo path simulation, Oper. Res. 56 2008, 3, 607–617] has a natural application to the evaluation of nested expectations {\mathbb{E}[g(\mathbb{E}[f(X,Y)|X])]}, where {f,g} are functions and {(X,Y)} a couple of independent random variables. Apart from the pricing of American-type derivatives, such computations arise in a large variety of risk valuations (VaR or CVaR of a portfolio, CVA), and in the assessment of margin costs for centrally cleared portfolios. In this work, we focus on the computation of initial margin. We analyze the properties of corresponding MLMC estimators, for which we provide results of asymptotic optimality; at the technical level, we have to deal with limited regularity of the outer function g (which might fail to be everywhere differentiable). Parallel to this, we investigate upper and lower bounds for nested expectations as above, in the spirit of primal-dual algorithms for stochastic control problems.

Sign in / Sign up

Export Citation Format

Share Document