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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,... 

Let G be a simple graph. An eigenvalue of G, is a main eigenvalue if the corresponding eigenspace has a non-orthogonal eigenvector to the all-one vector j. A signed graph is a graph with a sign to each edge. If in the adjacency matrix of background graph change elements that corresponded by -1, set -1 and in the other elements don’t make any change, then we reach the sign matrix of a signed graph. By an eigenvalue of a signed graph, we mean an eigenvalue of its sign matrix. In this research, we study main eigenvalues of graphs and signed graphs. At first, we present the necessary and sufficient conditions for any graph which has exactly m main eigenvalues. Then, by introducing and creating... 

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... 

In 1975 a Chemist Milan Randić proposed a concept named Randić index which is defined as follows: This index is generalized by replacing any real number α with which is called the general Randić index. Let G be a graph of order n. Erdős and Bollobás showed the lower bound for Randić index, Also, an upper bound for Randić index is n/2. In 2018 Suil O and Yongtang Shi proved a lower bound with minimum and maximum degree of a graph. They have shown for graph G we have, R(G) Also, a relation between Randić index and the energy of the graph has found. Indeed, it was proved that E(G) ⩾ 2R(G), where E(G) is the energy of graph. Many important bounds related to graph parameters for Randić index... 

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... 

All known examples of homogeneous Einstein spaces with negative scalar curvature (non compact) are isometric to standard Einstein solvmanifolds . we prove that any nilpotent Lie algebra having a codimension-one abelian ideal is the nilradical of a rank –one Einstein solvmanifold . In other words this nilpotent Lie algebra admits a rank-one solvable extension which can be endowed with an Einstein left invariant Riemannian metric . also a curve of pairwise non-isometric 8-dimensional rank-one Einstein solvmanifold is given .
 

In the recent years, the role of combinatorics and graph theory have grown in the progress of computer sciences. For instance, the circulant graphs have applications in design of interconnection networks and the graphs with integer eigenvalues are applied in modelling quantum spin networks supporting the perfect state transfer. The circulant graphs with integer eigenvalues also found applications in molecular graph energy. In 2006, it was shown that an n-vertex circulant graph G has integer eigenvalues if G=Cay(Zn; T ) or G= Cay(Zn; T)∪Cay(Zn;U(Zn)), where T Z(Zn). The Cayley graph Cay(Zn;U(Zn)) is known as the unitary Cayley graph. Fuchs defined the unitary Cayely graph of a commutative... 

Analytic hierarchy process (AHP) is a structured technique for organizing and analyzing complex decision. This process is used around the world in a wide variety of decision situation, in fields such as government, business, industry, healthcare, and education. In this method, decision problem first is decomposed into a hierarchy of more easily comprehended sub-problems, each of which can be analyzed independently. Once the hierarchy is built, the decision makers systematically evaluate its various elements by comparing them to one another two at a time, with respect to their impact on an element above them in the hierarchy. In this thesis we will study the hierarchy process and also study a... 

The current study is dedicated to free vibration analysis of a thin cylindrical shell with an oblique end. To this end, governing equations of motion and the corresponding boundary conditions are derived using Hamiltonian formulations. The differential form of equations is obtained by applying by part integration to the integral form of equations of motion. Equations of motions have been solved by Galerkin method for two different kinds of boundary conditions. Convergence process for different kinds of conditions has been done and results compared with papers. The effect of different parameters such as, length of cylinder, oblique angle, thickness etc. on the fundamental frequencies has been... 

Energy of graphs first defined by Ivan Gutman in 1978[1]. Let G be a graph with (0,1)-adjacency matrix A and let λ_1≥⋯≥λ_n be eigenvalues of A. Grap h energy is defined as the sum of absolute values of the eigenvalues of A and is shown by ɛ(G). let H_1,…,H_k be the vertex-disjoint induced subgraphs of graph G, it is proved that energy of G is at least equal to the sum of energy of H_i subgraphs, where the summation is over i. Also by partitioning edges of G to L_1,…,L_k subgraphs, energy of G is at most equal to sum of energy of L_i subgraphs , where the summation is over i. In this thesi s we study energy of graphs, specially... 

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... 

A fullerene graph is a cubic and 3-connected plan graph that has exactly 12 faces of size5 and other faces of size 6, which can be regarded as the molecular graph of fullerene.In the irst part of this thesis we study some important deinitions and theorems whichused in the other parts.A matching of a graph G is a set M of edges of G such that no two edges of M sharean end-vertex; further a matching M of G is perfect if any vertex of G is incident with anedge of M. A matching M of G is maximum if |M| ? |N| for any other matching N in G. Amatching M is maximal if it is not a proper subset of some other matching in G. Obviously,any maximum matching in G is also a maximal matching. An... 

The purpose of this thesis is to find upper bounds for the eigenvalues of natural operators acting on functions on a compact Riemannian man-ifold (M, g) such as the Laplace–Beltrami operator and Laplace-type operators. In the case of the Laplace–Beltrami operator, two aspects are investigated:
The first aspect is to study relationships between the intrinsic geometry and eigenvalues of the Laplace–Beltrami operator. In this regard, we obtain upper bounds depending only on the dimension and a conformal invariant called min-conformal volume. Asymptotically, these bounds are consistent with the Weyl law. They improve previous results by Korevaar and Yang and Yau. The proof relies on the... 

The current study is dedicated to free vibration and Aeroelastic Stability Analysis of Truncated Conical Panels in Supersonic Flows. Governing equations of motion and the corresponding boundary conditions are derived using Hamiltonian formulations. The aeroelastic stability problem is formulated based on first-order shear deformation theory as well as classical shell theory with the linearized first-order piston theory for aerodynamic loading and solved using Galerkin method. The flutter boundaries are obtained for truncated conical shells with different semi-vertex cone angles, different subtended angles, and different thickness