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algebraic-connectivity
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Topics of Stochastic Algebraic Topology
, M.Sc. Thesis Sharif University of Technology ; Jafari, Amir (Supervisor)
Abstract
Stochastic Algebraic Topology studies stochastic or less known spaces that depend on many random variables from the perspective of algebra. Examples of such spaces included random graphs of Erdos and Renyi and random simplicial spaces of higher dimension . In this thesis , we study algebraic aspects of these spaces such as homotopy groups and homology groups from probabilistic point of view , when the number of vertices of these spaces tend to infinity . The study of these spaces have a lot of applications, since the data for a given space can be incomplete . They are also used in big system structures. We have given a collection of new results in stochastic algebraic topology of two...
The algebraic connectivity of a graph and its complement
, Article Linear Algebra and Its Applications ; Volume 555 , 2018 , Pages 157-162 ; 00243795 (ISSN) ; Akbari, S ; Moghaddamzadeh, M. J ; Mohar, B ; Sharif University of Technology
Elsevier Inc
2018
Abstract
For a graph G, let λ2(G) denote its second smallest Laplacian eigenvalue. It was conjectured that λ2(G)+λ2(G‾)≥1, where G‾ is the complement of G. In this paper, it is shown that max{λ2(G),λ2(G‾)}≥[Formula presented]. © 2018 Elsevier Inc
Data Transmission Networks and Cayley Graphs
, M.Sc. Thesis Sharif University of Technology ; Ardeshir, Mohammad (Supervisor)
Abstract
A data transmission network constitutes of finite number of data processing stations which are connected together through data transmission lines. Each data transmission network corresponds to a graph which is called network’s graph. Obviously there are meaningful relations between structural properties of network’s graph and efficiency of network. In this thesis we investigate two aspects of network’s graph. The first subject is Cayley graphs. It has been shown that Cayley graphs are good models for designing data transmission networks. In this thesis we have surveyed some structural properties of Cayley graphs. The second subject is isoperimetric number. The isoperimetric number is a good...
Connectedness of users-items networks and recommender systems
, Article Applied Mathematics and Computation ; Vol. 243 , 2014 , Pages 578-584 ; ISSN: 00963003 ; Jalili, M ; Sharif University of Technology
Abstract
Recommender systems have become an important issue in network science. Collaborative filtering and its variants are the most widely used approaches for building recommender systems, which have received great attention in both academia and industry. In this paper, we studied the relationship between recommender systems and connectivity of users-items bipartite network. This results in a novel recommendation algorithm. In our method recommended items are selected based on the eigenvector corresponding to the algebraic connectivity of the graph - the second smallest eigenvalue of the Laplacian matrix. Since recommending an item to a user equals to adding a new link to the users-items bipartite...
A lower bound for algebraic connectivity based on the connection-graph- stability method
, Article Linear Algebra and Its Applications ; Volume 435, Issue 1 , Sep , 2011 , Pages 186-192 ; 00243795 (ISSN) ; Jalili, M ; Hasler, M ; Sharif University of Technology
2011
Abstract
This paper introduces the connection-graph-stability method and uses it to establish a new lower bound on the algebraic connectivity of graphs (the second smallest eigenvalue of the Laplacian matrix of the graph) that is sharper than the previously published bounds. The connection-graph-stability score for each edge is defined as the sum of the lengths of the shortest paths making use of that edge. We prove that the algebraic connectivity of the graph is bounded below by the size of the graph divided by the maximum connection-graph-stability score assigned to the edges
Trees with a large Laplacian eigenvalue multiplicity
, Article Linear Algebra and Its Applications ; Volume 586 , 2020 , Pages 262-273 ; van Dam, E. R ; Fakharan, M. H ; Sharif University of Technology
Elsevier Inc
2020
Abstract
In this paper, we study the multiplicity of the Laplacian eigenvalues of trees. It is known that for trees, integer Laplacian eigenvalues larger than 1 are simple and also the multiplicity of Laplacian eigenvalue 1 has been well studied before. Here we consider the multiplicities of the other (non-integral) Laplacian eigenvalues. We give an upper bound and determine the trees of order n that have a multiplicity that is close to the upper bound [Formula presented], and emphasize the particular role of the algebraic connectivity. © 2019 Elsevier Inc
Application of Algebraic Topology in Learning and Data Science
, M.Sc. Thesis Sharif University of Technology ; Jafari, Amir (Supervisor)
Abstract
This thesis has provided an overview of the theoretical justification for the discrete Morse theory، persistent homology، and applications. Once the primary definitions of complexes have been established in the first chapter، we have explored the Discrete morse theory in detail in the second chapter. Then it is used to prove that disconnectivity is an eva- sive feature of graphs. Furthermore، some theorems related to discrete morse theory have been used to reduce the time complexity of ho- mology algorithms. Next chapter، we try to establish the foundation of persistent homology، and then we explore stability theorems of persis- tent homology. Finally، an application chapter has been added،...
A graph weighting method for reducing consensus time in random geographical networks
, Article 24th IEEE International Conference on Advanced Information Networking and Applications Workshops, WAINA 2010, 20 April 2010 through 23 April 2010, Perth ; 2010 , Pages 317-322 ; 9780769540191 (ISBN) ; Sharif University of Technology
2010
Abstract
Sensor networks are increasingly employed in many applications ranging from environmental to military cases. The network topology used in many sensor network applications has a kind of geographical structure. A graph weighting method for reducing consensus time in random geographical networks is proposed in this paper. We consider a method based on the mutually coupled oscillators for providing general consensus in the network. In this way, one can relate the consensus time to the properties of the Laplacian matrix of the connection graph, i.e. to the second smallest eigenvalue (algebraic connectivity). Our weighting algorithm is based on the node and edge between centrality measures. The...