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Let G be a graph of order n. The signless Laplacian matrix or Q-matrix of G is Q(G)=D(G)+A(G), where A(G) is the adjacency matrix of G and D(G) is diagonal degree matrix of G. The signless Laplacian characteristic polynomial or Q-polinomial of G is QG(x)=det(xI-Q(G)) and its roots are called signless Laplacian eigenvalues or Q-eigenvalues of G. If R is vertex-degree incidence matrix of G, then Q(G)=RRt. So Q(G) is a positive semi-definite matrix, i.e. its eigenvalues are none-negative. Let q1(G)≥q2(G)≥…≥qn(G) denote the signless Laplacian eigenvalues of G. A theory in which graphs are studied by means of eigenvalues of Q(G) is called signless Laplaciian theory or Q-theory.In this research,... 

Density matrix of graphs as defined -for the first time- in [S. Braunstein and et al. The laplacian of a graph as a density matrix, Annals of Combinatorics, (2006)], is obtained through dividing the Laplacian matrix by the degree sum. This matrix is also semi-positive and trace one. Therefore one may talk about the Von Neumann entropy of this matrix. In [F. Passerini, S. Severini. Quantifying complexity in networks: The Von Neumann entropy. IJATS, (2009)], authors have claimed that this quantity can be consisered as a measure of regularity. Here, using a geometric interpretation of Von Neumann entropy, expresed in [G. Mitchison, R. Jozsa, Towards a geometrical interpretation of quantum... 

A signed graph is a pair like (G; ), where G is the underlying graph and : E(G) ! f1; +1g is a sign function on the edges of G. Here, we study the spectral determination problem for signed n-cycles (Cn; ) with respect to the adjacency spectrum and the Laplacian spectrum. In particular we prove that signed odd cycles and unbalanced even cycles are uniquely determined by their Laplacian spectrums, but balanced even cycles are not, and we find all L-cospectral mates for them. Moreover, signed odd cycles are uniquely determined by their spectrums but the signed even cycles, (except (C4;) and (C4; +)), are not and we find almost all cospectral mates for them. A mixed graph is obtained from a... 

Recently, there has been a surge of research activities in the area of networks in the systems and control community. One foundational class of questions on networked systems pertains to their controllability. While there are classical tests to check the controllability of linear time-invariant (LTI) systems, their applications to large-scale networks is numerically infeasible. Indeed, finding a minimum cardinality set of input nodes that ensure the controllability of a network is NP-hard. Moreover, when the interaction strengths along the edges of a network are unknown, the classical controllability test cannot be applied. To overcome these issues, an alternative set of approaches involves... 

The spectrum of a graph is related to many important combinatorial parameters. Let (G), ′(G) be the maximum number of edge-disjoint spanning trees and edge-connectivity of a graph G,respectively. Motivated by a question of Seymour on the relationship between eigenvalues of a graph G and bounds of (G), we use eigenvalue interlacing for quotient matrix associated to graph to get the relationship between eigenvalues of a graph and bounds of (G) and ′(G). We also study the relationship between eigenvalues and bounds of (G) and ′(G) in a multigraph G. In the first chapter we prove eigenvalue interlacing and give several applications of it for obtaining bounds for characteristic numbers of... 

The running time of almost every algorithm in the graph theory depend on the number of edges. Thus, these algorithms are often too slow when the input graphs are dense. Therefore, it is useful to reduce the number of edges by sparsification. In fact, sparsification is the task of approximating a graph G = (V;E) by another graph ~G = (V; ~E) so that ~E E (j~Ej jEj) and ~G maintain a main prefixed property of G. Depending on these properties several notions of graph sparsification have been proposed. In this thesis we study a notion of sparsification that is called spectral sparsification which is based on the contributions of Daniel A. Spielman et.al..In this notion of sparsification... 

In this thesis, we study the treewidth of social networks. The importance of studding treewidth is for two reasons. The first is that for the graph with bounded treewidth, many optimization problems that are NP-hard in general, can be solved in polynomial or even linear time. The second is that the high value of treewidth in a graph, reflects some high degree of connectivity and robustness, which is an essential factor in designing many networks. But the problem is that determining the value of treewidth in a graph is NP-complete so, computing the treewidth of real complex networks is not feasible. We first review the related works and mention the weakness of the past works, then introduce a... 

Let G be a graph with adjacency matrix A and Δ be a diagonal matrix whose diagonal entries are the degree sequence of G. Then the matrices L = Δ− A and Q = Δ+A are called Laplacian matrix and signless Laplacian matrix of G, respectively. The eigenvalues of A, L, and Q are arranged decreasingly and denoted by λ1 ≥ · · · ≥ λn, μ1 ≥ · · · ≥ μn ≥ 0, and q1 ≥ · · · ≥ qn ≥ 0, respectively. The energy of a graph G is defined as E(G) :=
n
i=1
|λi|.
Furthermore, the incidence energy, the signed incidence energy, and the H¨uckel energy of G are
defined as
IE(G) :=
n
i=1
√
qi, LE(G) :=
n
i=1
√
μi, HE(G) :=

2
r
i=1 λi, n=... 

In this thesis we investigate the spectrum of the Laplacian matrix of a graph. Although its use dates back to Kirchhoff, most of the major results are much more recent. The first chapter of this thesis is devoted to the integral Laplacian eigenvalues of graphs. In Section 2, particular attention is given to multiplicities of integer eigenvalues and to the effect on the spectrum of various modifications. In Section 3, the Laplacian integral graphs are investigated. The Section 4 relates the degree sequence and the Laplacian spectrum through majorization.The second chapter presents the result on permanent of the Laplacian matrix of graphs and permanental roots. In Section 2, we investigate... 

In this thesis, we study the multiplicity of the Laplacian eigenvalues of trees. It is known that for trees, integer Laplacian eigenvalues larger than 1 are simple. Here we consider the multiplicities of the other (non- integral) Laplacian eigenvalues.We provide an upper bound and determine the trees of order n that have a multiplicity that is close to the upper bound (n-3)/2 , and emphasize the particular role of the algebraic connectivity.In continuation, let G be a graph and I be an interval. We present bounds for the number m_G I of Laplacian eigenvalues in I in terms of structural parameters of G. In particular, we show that m_G (n-α(G),n]≤ n-α(G) and m_G (n-d(G)+3,n]≤ n-d(G)-1, where... 

A signed graph is a graph with a sign attached to each edge. This article extends some fundamental concepts of the Laplacian matrices from graphs to signed graphs.In particular, the largest Laplacian eigenvalue of a signed graph is investigated,which generalizes the corresponding results on the largest Laplacian eigenvalue of a graph.It is proved that (C2n+1; +) is uniquely determined by its Laplacian spectrum (or is DLS), where (C2n+1; +) is a signed cycle in which all edges have positive sign. On the other hand, we determine all Laplacian cospectral mates of (C2n; +) and hence (C2n; +) is not DLS. Also, we show that for every positive integer n, (Cn;) is DLS. Then, we study the spectrum of... 

In chis chcsis we consider a work by Dmitry Dolg:opyat and Raphac) Kriko­ rian (on Simulcanoous Lincariwtion of Diffcomorphisms of Sphere) and a recent preprint by Jonathan Dc\Vitt (Simulrn.ncous Lincarization of Diffco­ morphisms of Isotropic Manifolds). Suppasc /1,h,...,J.. arc pcrturbacions of rotations R1, R2 , ...,Rn of the sphere which generate SOd+I (d 2). \\Tc c". plian how it can be shown that all /; can besimn)tancously conjugated to some new rotations when all the Lyapunov cxponcms of the rcndom walk a.,:;.,sociatc to {/;}arc zero. This rcsn1t is applied to obtain stable ergodicity when d is even.The idea is co translate al1 the information co a linear setting and approach chc... 

Partial discharges that occur in a transformer insulation, generate current pulses. If these pulses be recorded, they can be used for transformer insulation condition assessment. Through processing of these recorded partial discharge signals, the PRPD patterns are generated and used to identify the source type of partial discharge defect. If multiple partial discharge defects exist in a transformer insulation, the related PRPD pattern, doesn’t look like any PRPD patterns of single defects. In this case, we need in the first step to discriminate the partial discharge signals stemmed from all the existing multiple partial discharge sources. To simulate the occurrence of multiple partial... 

Graph partitioning, or graph clustering, is an essential researa problem in many areas. In this thesis, we focus on the partitioning problem of unweighted undirected graph, that is, graphs for which the weight of all edges is 1. We will investigate spectral methods for solving the graph partitioning and we compare them. In addition to theoretical analysis,We also implement some of spectral algorithms in matlab and apply them on standard graph data sets. Finally, the experimental
results obtained are offering  

In this thesis we have proven A variational representation for positive functionals of a Hilbert space valued Wiener process (W(.)).This representation is then used to prove a laplace principle for the family{G(W(.))}>0 where G is an appropriate family of measurable maps from the Wiener space to some Polish space  

Medical imaging is of interest because of information that will provide for doctors and registration is inevitable when we need to compare two or more images, taken from a subject at different times or with different sensors or when comparing two or more subjects together. Registration methods can be categorized in two major groups; methods based on feature and methods based on intensity. Methods in first group have three steps in common: feature extraction, finding matches and transform estimation. In second group it’s important to define a similarity measure and find the transform that minimizes this measure.
Manifold learning algorithms are mostly used as a dimensionality reduction...