282 results on '"Adjacency matrix"'
Search Results
2. Community Detection Through Optimal Density Contrast of Adjacency Matrix.
- Author
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Liang, Tianzhu and Szeto, Kwok Yip
- Subjects
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DATA analysis , *CLUSTER analysis (Statistics) , *TOPOLOGY , *GREEDOIDS , *MATRICES (Mathematics) , *MATHEMATICAL analysis - Abstract
Detecting communities in real world networks is an important problem for data analysis in science and engineering. By clustering nodes intelligently, a recursive algorithm is designed to detect community. Since the relabeling of nodes does not alter the topology of the network, the problem of community detection corresponds to the finding of a good labeling of nodes so that the adjacency matrix form blocks. By putting a fictitious interaction between nodes, the relabeling problem becomes one of energy minimization, where the total energy of the network is defined by putting interaction between the labels of nodes so that clustering nodes that are in the same community will decrease the total energy. A greedy method is used for the computation of minimum energy. The method shows efficient detection of community in artificial as well as real world network. The result is illustrated in a tree showing hierarchical structure of communities on the basis of sub-matrix density. Applications of the method to weighted and directed networks are discussed. [ABSTRACT FROM AUTHOR]
- Published
- 2011
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3. Cut-and-paste of quadriculated disks and arithmetic properties of the adjacency matrix
- Author
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Saldanha, Nicolau C. and Tomei, Carlos
- Subjects
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MATRICES (Mathematics) , *GEOMETRIC analysis , *DIMERS , *ARITHMETIC , *MATHEMATICAL analysis , *MATCHING theory - Abstract
Abstract: We define cut-and-paste, a construction which, given a quadriculated disk obtains a disjoint union of quadriculated disks of smaller total area. We provide two examples of the use of this procedure as a recursive step. Tilings of a disk receive a parity: we construct a perfect or near-perfect matching of tilings of opposite parities. Let be the black-to-white adjacency matrix: we factor , where and are lower and upper triangular matrices, is obtained from a larger identity matrix by removing rows and columns and all entries of , and are equal to , or . [Copyright &y& Elsevier]
- Published
- 2010
- Full Text
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4. Mutually touching infinite cylinders in the 3D world of lines
- Author
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Stanislaw Pikhitsa and Peter V. Pikhitsa
- Subjects
Physics ,General Mathematics ,Mathematical analysis ,52C17, 65H04, 57M99 ,Boundary (topology) ,Metric Geometry (math.MG) ,Graph theory ,Upper and lower bounds ,Cylinder (engine) ,law.invention ,Matrix (mathematics) ,Mathematics - Metric Geometry ,law ,Line (geometry) ,Chirality (mathematics) ,FOS: Mathematics ,Adjacency matrix - Abstract
Recently we gave arguments that only two unique topologically different configurations of 7 equal all mutually touching round cylinders (the configurations being mirror reflections of each other) are possible in 3D, although a whole world of configurations is possible already for round cylinders of arbitrary radii. It was found that as many as 9 round cylinders (all mutually touching) are possible in 3D while the upper bound for arbitrary cylinders was estimated to be not more than 14 under plausible arguments. Now by using the chirality and Ring matrices that we introduced earlier for the topological classification of line configurations, we have given arguments that the maximal number of mutually touching straight infinite cylinders of arbitrary cross-section (provided that its boundary is a smooth curve) in 3D cannot exceed 10. We generated numerically several configurations of 10 cylinders, restricting ourselves with elliptic cylinders. Configurations of 8 and 9 equal elliptic cylinders (all in mutually touching) are generated numerically as well. A possibility and restriction of continuous transformations from elliptic into round cylinder configurations are discussed. Some curious results concerning the properties of the chirality matrix (which coincides with Seidel's adjacency matrix important for the Graph theory) are presented., 27 pages, 10 figures
- Published
- 2019
5. Loewner’s Differential Equation and Spidernets
- Author
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Sebastian Schleißinger
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Differential equation ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Function (mathematics) ,Operator theory ,01 natural sciences ,Computational Mathematics ,Quantum probability ,Computational Theory and Mathematics ,Quantum process ,Product (mathematics) ,0103 physical sciences ,Point (geometry) ,010307 mathematical physics ,Adjacency matrix ,0101 mathematics ,Mathematics - Abstract
We regard Loewner’s differential equation with non-negative continuous driving functions (and more generally driving measures with non-negative support) from a quantum probability point of view and approximate the underlying quantum process by the adjacency matrices of growing graphs which arise from the comb product of certain spidernets. In the general case of an arbitrary continuous driving function we find a weaker approximation by comb products of graphs with weighted loops.
- Published
- 2019
6. On the Use of Discrete-continual Finite Element Method with Unstructured Meshes
- Author
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Pavel A. Akimov and Oleg A. Negrozov
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Mathematical analysis ,Piecewise ,Direct stiffness method ,Geometry ,General Medicine ,Adjacency matrix ,Mixed finite element method ,Single-entry matrix ,Finite element method ,Engineering(all) ,Stiffness matrix ,Mathematics ,Extended finite element method - Abstract
The field of application of discrete-continual finite element method comprises structures with regular (in particular, constant or piecewise constant) physical and geometrical parameters in some dimension (so-called “basic” direction (dimension)). Considering problems remain continual along “basic” direction while along other directions mesh approximation is performed. Algorithm of computing of local stiffness matrix of discrete-continual finite element with quadrangular cross-section and different integration schemes is presented. Constructing of global stiffness matrix with the use of adjacency matrix and several numerical samples are considered as well.
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- 2015
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7. Spectral statistics of lattice graph structured, non-uniform percolations
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Stephen Kruzick and Jose M. F. Moura
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Random graph ,Mathematical analysis ,Computer Science - Numerical Analysis ,020206 networking & telecommunications ,Numerical Analysis (math.NA) ,02 engineering and technology ,01 natural sciences ,010104 statistics & probability ,Integer matrix ,Matrix (mathematics) ,Graph energy ,FOS: Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Statistical physics ,Adjacency matrix ,0101 mathematics ,Laplacian matrix ,Random matrix ,Eigenvalues and eigenvectors ,Mathematics - Abstract
Design of filters for graph signal processing benefits from knowledge of the spectral decomposition of matrices that encode graphs, such as the adjacency matrix and the Laplacian matrix, used to define the shift operator. For shift matrices with real eigenvalues, which arise for symmetric graphs, the empirical spectral distribution captures the eigenvalue locations. Under realistic circumstances, stochastic influences often affect the network structure and, consequently, the shift matrix empirical spectral distribution. Nevertheless, deterministic functions may often be found to approximate the asymptotic behavior of empirical spectral distributions of random matrices. This paper uses stochastic canonical equation methods developed by Girko to derive such deterministic equivalent distributions for the empirical spectral distributions of random graphs formed by structured, non-uniform percolation of a D-dimensional lattice supergraph. Included simulations demonstrate the results for sample parameters., ICASSP 2017
- Published
- 2017
8. Symmetry, topology and the maximum number of mutually pairwise-touching infinite cylinders: configuration classification
- Author
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Peter V. Pikhitsa and Stanislaw Pikhitsa
- Subjects
02 engineering and technology ,Space (mathematics) ,01 natural sciences ,Cylinder (engine) ,law.invention ,Matrix (mathematics) ,law ,Seidel adjacency matrix ,0103 physical sciences ,chirality matrix ,Uniqueness ,mutually touching cylinders ,lcsh:Science ,Topology (chemistry) ,Mathematics ,010302 applied physics ,Multidisciplinary ,Mathematical analysis ,021001 nanoscience & nanotechnology ,Symmetry (physics) ,topological invariant ,Chirality (mathematics) ,lcsh:Q ,0210 nano-technology ,Research Article - Abstract
We provide a complete classification of possible configurations of mutually pairwise-touching infinite cylinders in Euclidian three-dimensional space. It turns out that there is a maximum number of such cylinders possible in three dimensions independently of the shape of the cylinder cross-sections. We give the explanation of the uniqueness of the non-trivial configuration of seven equal mutually touching round infinite cylinders found earlier. Some results obtained for the chirality matrix, which is equivalent to the Seidel adjacency matrix, may be found useful for the theory of graphs.
- Published
- 2016
9. UPPER AND LOWER BOUNDS FOR THE POWER OF EIGENVALUES IN SEIDEL MATRIX
- Author
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Ali Iranmanesh and Jalal Askari Farsangi
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Matrix differential equation ,Seidel adjacency matrix ,Mathematical analysis ,Applied mathematics ,Eigenvalue algorithm ,Conference matrix ,Upper and lower bounds ,Eigenvalues and eigenvectors ,Eigendecomposition of a matrix ,Mathematics ,Nonlinear programming - Published
- 2015
10. On the energy and spectral properties of the He matrix of hexagonal systems
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F M Bhatti, Kinkar Ch. Das, and Syed Ali Ahmed
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Combinatorics ,Graph energy ,Degree matrix ,Spectral radius ,General Mathematics ,Mathematical analysis ,Hexagonal number ,Incidence matrix ,Adjacency matrix ,Laplacian matrix ,Nonlinear Sciences::Pattern Formation and Solitons ,Eigenvalues and eigenvectors ,Mathematics - Abstract
The He matrix, put forward by He and He in 1989, is designed as a means for uniquely representing the structure of a hexagonal system (= benzenoid graph). Observing that the He matrix is just the adjacency matrix of a pertinently weighted inner dual of the respective hexagonal system, we establish a number of its spectral properties. Afterwards, we discuss the number of eigenvalues equal to zero of the He matrix of a hexagonal system. Moreover, we obtain a relation between the number of triangles and the eigenvalues of the He matrix of a hexagonal system. Finally, we present an upper bound on the He energy of hexagonal systems.
- Published
- 2013
11. Network Analysis via Partial Spectral Factorization and Gauss Quadrature
- Author
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Caterina Fenu, Giuseppe Rodriguez, David R. Martin, and Lothar Reichel
- Subjects
Applied Mathematics ,Mathematical analysis ,Low-rank approximation ,Spectral theorem ,Upper and lower bounds ,Exponential function ,Combinatorics ,Computational Mathematics ,Nonlinear system ,symbols.namesake ,symbols ,Gaussian quadrature ,Adjacency matrix ,Network analysis ,Mathematics - Abstract
Large-scale networks arise in many applications. It is often of interest to be able to identify the most important nodes of a network or to ascertain the ease of traveling between nodes. These and related quantities can be determined by evaluating expressions of the form $\mathbf{u}^Tf(A)\mathbf{w}$, where $A$ is the adjacency matrix that represents the graph of the network, $f$ is a nonlinear function, such as the exponential function, and $\mathbf{u}$ and $\mathbf{w}$ are vectors, for instance, axis vectors. This paper describes a novel technique for determining upper and lower bounds for expressions $\mathbf{u}^Tf(A)\mathbf{w}$ when $A$ is symmetric and bounds for many vectors $\mathbf{u}$ and $\mathbf{w}$ are desired. The bounds are computed by first evaluating a low-rank approximation of $A$, which is used to determine rough bounds for the desired quantities for all nodes. These rough bounds indicate for which vectors $\mathbf{u}$ and $\mathbf{w}$ more accurate bounds should be computed with the aid ...
- Published
- 2013
12. On two conjectures of Randić index and the largest signless Laplacian eigenvalue of graphs.
- Author
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Deng, Hanyuan, Balachandran, S., and Ayyaswamy, S.K.
- Subjects
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EIGENVALUES , *GRAPH theory , *LAPLACIAN matrices , *HARMONIC analysis (Mathematics) , *MATHEMATICAL proofs , *MATHEMATICAL analysis - Abstract
Abstract: The Randić index R of a graph G is defined as the sum of over all edges of G, where denotes the degree of a vertex in G. is the largest eigenvalue of the signless Laplacian matrix of G, where D is the diagonal matrix with degrees of the vertices on the main diagonal and A is the adjacency matrix of G. Hansen and Lucas [18] conjectured (1) and equality holds for and (2) with equality if and only if for and for , respectively. In this paper, we prove the conjecture (1) and obtain a result very close to the conjecture (2). Moreover, we give some results relating harmonic index and the largest eigenvalue of the adjacency matrix. [Copyright &y& Elsevier]
- Published
- 2014
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13. Smith forms for adjacency matrices of circulant graphs.
- Author
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Williams, Gerald
- Subjects
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GRAPH theory , *PATHS & cycles in graph theory , *CIRCULANT matrices , *ABELIAN groups , *PROOF theory , *MATHEMATICAL analysis - Abstract
Abstract: We calculate the Smith normal form of the adjacency matrix of each of the following graphs or their complements (or both): complete graph, cycle graph, square of the cycle, power graph of the cycle, distance matrix graph of cycle, Andrásfai graph, Doob graph, cocktail party graph, crown graph, prism graph, Möbius ladder. The proofs operate by finding the abelianization of a cyclically presented group whose relation matrix is column equivalent to the required adjacency matrix. [Copyright &y& Elsevier]
- Published
- 2014
- Full Text
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14. Reference free structure determination through eigenvectors of center of mass operators
- Author
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Fred J. Sigworth, Amit Singer, Ronald R. Coifman, and Yoel Shkolnisky
- Subjects
Unit sphere ,Geodesic ,Computer science ,Applied Mathematics ,Mathematical analysis ,010103 numerical & computational mathematics ,02 engineering and technology ,Radial line ,01 natural sciences ,Article ,symbols.namesake ,Fourier transform ,Iterative refinement ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,Graph (abstract data type) ,020201 artificial intelligence & image processing ,Adjacency matrix ,0101 mathematics ,Algorithm ,Eigenvalues and eigenvectors - Abstract
Recovering the three-dimensional structure of molecules is important for understanding their functionality. We describe a spectral graph algorithm for reconstructing the three-dimensional structure of molecules from their cryo-electron microscopy images taken at random unknown orientations. We first identify a one-to-one correspondence between radial lines in three-dimensional Fourier space of the molecule and points on the unit sphere. The problem is then reduced to determining the coordinates of points on the sphere given a subset of their pairwise geodesic distances. To recover those coordinates, we exploit the special geometry of the problem, as rendered by the Fourier projection–slice theorem, to construct a weighted graph whose vertices are the radial Fourier lines and whose edges are linked using the common line property. The graph organizes the radial lines on the sphere in a global manner that reveals the acquisition direction of each image. This organization is derived from a global computation of a few eigenvectors of the graph's sparse adjacency matrix. Once the directions are obtained, the molecule can be reconstructed using classical tomography methods. The presented algorithm is direct (as opposed to iterative refinement schemes), does not require any prior model for the reconstructed object, and is shown to have favorable computational and numerical properties. Moreover, the algorithm does not impose any assumption on the distribution of the projection orientations. Physically, this means that the algorithm is applicable to molecules that have unknown spatial preference.
- Published
- 2010
15. Relation between signless Laplacian energy, energy of graph and its line graph.
- Author
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Das, Kinkar Ch. and Mojallal, Seyed Ahmad
- Subjects
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LAPLACIAN matrices , *GRAPH theory , *ABSOLUTE value , *MATHEMATICAL bounds , *MATHEMATICAL analysis , *NUMERICAL analysis - Abstract
The energy of a simple graph G , E ( G ) , is the sum of the absolute values of the eigenvalues of its adjacency matrix. The energy of line graph and the signless Laplacian energy of graph G are denoted by E ( L G ) ( L G is the line graph of G ) and LE + ( G ) , respectively. In this paper we obtain a relation between E ( L G ) and LE + ( G ) of graph G . From this relation we characterize all the graphs satisfying E ( L G ) = LE + ( G ) + 4 m n − 4 . We also present a relation between E ( G ) and E ( L G ) . Moreover, we give an upper bound on E ( L G ) of graph G and characterize the extremal graphs. [ABSTRACT FROM AUTHOR]
- Published
- 2016
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16. A note on connected bipartite graphs of fixed order and size with maximal index.
- Author
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Petrović, Miroslav and Simić, Slobodan K.
- Subjects
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BIPARTITE graphs , *MAXIMAL subgroups , *EIGENVALUES , *MATRICES (Mathematics) , *GRAPH theory , *MATHEMATICAL analysis - Abstract
In this paper the unique graph with maximal index (i.e. the largest eigenvalue of the adjacency matrix) is identified among all connected bipartite graphs of order n and size n + k , under the assumption that k ≥ 0 and n ≥ k + 5 . [ABSTRACT FROM AUTHOR]
- Published
- 2015
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17. A characterization of graphs with rank 5
- Author
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Chang, Gerard J., Huang, Liang-Hao, and Yeh, Hong-Gwa
- Subjects
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GRAPH theory , *MATRICES (Mathematics) , *GRAPH connectivity , *MATHEMATICS , *MATHEMATICAL analysis - Abstract
Abstract: The rank of a graph G is defined to be the rank of its adjacency matrix. In this paper, we consider the following problem: what is the structure of a connected graph G with rank 5? or equivalently, what is the structure of a connected n-vertex graph G whose adjacency matrix has nullity ? In this paper, we completely characterize connected graphs G whose adjacency matrix has rank 5. [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
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18. A characterization of graphs with rank 4
- Author
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Chang, Gerard J., Huang, Liang-Hao, and Yeh, Hong-Gwa
- Subjects
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GRAPH theory , *MATHEMATICAL analysis , *MATRICES (Mathematics) , *SPECTRAL theory , *ALGEBRA , *ABSTRACT algebra - Abstract
Abstract: The rank of a graph G is defined to be the rank of its adjacency matrix. In this paper, we consider the following problem: What is the structure of a connected graph with rank 4? This question has not yet been fully answered in the literature, and only some partial results are known. In this paper we resolve this question by completely characterizing graphs G whose adjacency matrix has rank 4. [Copyright &y& Elsevier]
- Published
- 2011
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19. Adjacency matrices of probe interval graphs
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Ghosh, Shamik, Podder, Maitry, and Sen, Malay K.
- Subjects
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BIPARTITE graphs , *INTERVAL analysis , *MATHEMATICAL proofs , *DIMENSIONAL analysis , *MATHEMATICAL symmetry , *MATHEMATICAL analysis - Abstract
Abstract: In this paper we obtain several characterizations of the adjacency matrix of a probe interval graph. In course of this study we describe an easy method of obtaining interval representation of an interval bigraph from its adjacency matrix. Finally, we note that if we add a loop at every probe vertex of a probe interval graph, then the Ferrers dimension of the corresponding symmetric bipartite graph is at most 3. [ABSTRACT FROM AUTHOR]
- Published
- 2010
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20. Cayley digraphs with normal adjacency matrices
- Author
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Lyubshin, David S. and Savchenko, Sergey V.
- Subjects
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CAYLEY graphs , *DIRECTED graphs , *MATRICES (Mathematics) , *ABELIAN groups , *QUATERNIONS , *MATHEMATICAL analysis - Abstract
Abstract: In this paper we give a criterion for the adjacency matrix of a Cayley digraph to be normal in terms of the Cayley subset . It is shown with the use of this result that the adjacency matrix of every Cayley digraph on a finite group is normal iff is either abelian or has the form for some non-negative integer , where is the quaternion group and is the abelian group of order and exponent 2. [Copyright &y& Elsevier]
- Published
- 2009
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21. Non-bipartite graphs of fixed order and size that minimize the least eigenvalue.
- Author
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Jovović, Ivana, Koledin, Tamara, and Stanić, Zoran
- Subjects
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BIPARTITE graphs , *EIGENVALUES , *SET theory , *MATHEMATICAL constants , *MATHEMATICAL analysis - Abstract
In this paper we determine the unique graph with minimal least eigenvalue (of the adjacency matrix) within the set of connected graphs of fixed order n and size m , whenever m = ⌈ n 2 ⌉ ⌊ n 2 ⌋ + a , where a is a fixed integral constant in [ 1 , ⌈ n 2 ⌉ − 1 ] . [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
22. On the multiplicity of the adjacency eigenvalues of graphs.
- Author
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Bahmani, Asghar and Kiani, Dariush
- Subjects
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MULTIPLICITY (Mathematics) , *GRAPH theory , *EIGENVALUES , *MATRICES (Mathematics) , *MATHEMATICAL analysis - Abstract
Let G be a simple graph with the adjacency matrix A ( G ) . A well-known result of Cvetković and Gutman states that removing a pendant vertex and its neighbour, does not change the nullity of the graph. We generalize this theorem and some other theorems similar to it for an arbitrary eigenvalue of a graph. Also, for an arbitrary eigenvalue λ , we use a corresponding star set to delete some subgraphs and to determine the multiplicity of λ . We use star sets to find some Parter–Wiener vertices, in trees. We state our results for real symmetric matrices (and so for weighted graphs) and as a corollary we use them for simple graphs. Furthermore, we use these methods to obtain some results for λ = 0 , ± 1 . [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
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23. The Least Eigenvalue of Graphs whose Complements Are Uni- cyclic.
- Author
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Wang, Yi, Fan, Yi-Zheng, Li, Xiao-Xin, and Zhang, Fei-Fei
- Subjects
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EIGENVALUES , *GRAPH theory , *SET theory , *GRAPH connectivity , *MATHEMATICAL analysis - Abstract
A graph in a certain graph class is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum among all graphs in that class. Bell et al. have identified a subclass within the connected graphs of order n and size m in which minimizing graphs belong (the complements of such graphs are either disconnected or contain a clique of size n/2 ). In this paper we discuss the minimizing graphs of a special class of graphs of order n whose complements are connected and contains exactly one cycle (namely the class Ucn of graphs whose complements are unicyclic), and characterize the unique minimizing graph in Ucn when n ≥ 20. [ABSTRACT FROM AUTHOR]
- Published
- 2015
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24. State transfer and star complements in graphs.
- Author
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Jiang Zhou and Changjiang Bu
- Subjects
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GRAPH theory , *EIGENVALUES , *MULTIPLICITY (Mathematics) , *OPERATOR theory , *MATRIX groups , *MATHEMATICAL analysis - Abstract
Let G be a graph with adjacency matrix A, and let H(t)=exp(itA). For an eigenvalue μ of A with multiplicity k, a star set for μ in G is a vertex set X of G such that |X|=k and the induced subgraph G-X does not have μ as an eigenvalue. G is said to have perfect state transfer from the vertex u to the vertex v if there is a time τ such that |H(τ)u,v|=1. The unitary operator H(t) has important applications in the transfer of quantum information. In this paper, we give an expression of H(t). For a star set X of graph G, perfect state transfer does not occur between any two vertices in X. We also give some results for the existence of perfect state transfer in a graph. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
25. TWO GRAPHS WITH A COMMON EDGE.
- Author
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BADURA, LIDIA
- Subjects
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GRAPH theory , *PATHS & cycles in graph theory , *DETERMINANTS (Mathematics) , *SUBGRAPHS , *MATRICES (Mathematics) , *MATHEMATICAL analysis - Abstract
Let G = G1 ∪ G2 be the sum of two simple graphs G1, G2 having a common edge or G = G1 ∪ e1 ∪ e2 ∪ G2 be the sum of two simple disjoint graphs G1,G2 connected by two edges e1 and e2 which form a cycle C4 inside G. We give a method of computing the determinant detA(G) of the adjacency matrix of G by reducing the calculation of the determinant to certain subgraphs of G1 and G2. To show the scope and effectiveness of our method we give some examples [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
26. Chronographic Imprint of Age-Induced Alterations in Heart Rate Dynamical Organization
- Author
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Marta Żarczyńska-Buchowiecka, Dorota Wejer, Agnieszka Kaczkowska, Danuta Makowiec, and Zbigniew R. Struzik
- Subjects
lcsh:QP1-981 ,Series (mathematics) ,Computer science ,Physiology ,entropy rate ,Mathematical analysis ,aging ,computer.software_genre ,lcsh:Physiology ,deceleration capacity ,Physiology (medical) ,Contour line ,Heart rate ,Line (geometry) ,short-term heart rate variability ,Transition matrices ,Adjacency matrix ,Data mining ,Maxima ,computer ,Cardiac vagal control ,Entropy rate ,Original Research - Abstract
Beat-to-beat changes in the heart period are transformed into a network of increments bet-ween subsequent RR-intervals, which enables graphical descriptions of short-term heart period variability. Three types of such descriptions are considered: (1) network graphs arising from a set of vertices and directed edges, (2) contour plots of adjacency matrices A, representing the networks and transition matrices T, resulting from A, and (3) vector plots of gradients of the matrices A and T. Two indices are considered which summarize properties of A and T: the approximate deceleration capacity and the entropy rate. The method, applied to time series of nocturnal RR-intervals recorded from healthy subjects of different ages, reveals important aspect of changes in the autonomic activity caused by biological aging. Independently of the subject's age, following accelerations a pendulum-like dynamics appears. With decelerations, this dynamics develops in line with the subject's age. This aging transition can be graphically visualized by vectors connecting the maxima of the transition probabilities of~T, which, metaphorically, resemble a chronometer or the hands of a clock.
- Published
- 2015
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27. Universality in the spectral and eigenfunction properties of random networks
- Author
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A. Alcazar-Lopez, A. J. Martinez-Mendoza, Francisco A. Rodrigues, J. A. Méndez-Bermúdez, and Thomas K. Dm. Peron
- Subjects
Mathematical analysis ,Diagonal ,PROBABILIDADE ,FOS: Physical sciences ,Network size ,Disordered Systems and Neural Networks (cond-mat.dis-nn) ,Condensed Matter - Disordered Systems and Neural Networks ,Eigenfunction ,Network connectivity ,Universality (dynamical systems) ,Statistical physics ,Adjacency matrix ,Randomness ,Mathematics ,Network model - Abstract
By the use of extensive numerical simulations we show that the nearest-neighbor energy level spacing distribution $P(s)$ and the entropic eigenfunction localization length of the adjacency matrices of Erd\H{o}s-R\'enyi (ER) {\it fully} random networks are universal for fixed average degree $\xi\equiv \alpha N$ ($\alpha$ and $N$ being the average network connectivity and the network size, respectively). We also demonstrate that Brody distribution characterizes well $P(s)$ in the transition from $\alpha=0$, when the vertices in the network are isolated, to $\alpha=1$, when the network is fully connected. Moreover, we explore the validity of our findings when relaxing the randomness of our network model and show that, in contrast to standard ER networks, ER networks with {\it diagonal disorder} also show universality. Finally, we also discuss the spectral and eigenfunction properties of small-world networks., Comment: 11 pages, 9 figures
- Published
- 2015
28. On oriented graphs whose skew spectral radii do not exceed 2.
- Author
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Xu, Guang-Hui and Gong, Shi-Cai
- Subjects
- *
GRAPH theory , *SPECTRAL geometry , *MATRICES (Mathematics) , *EIGENVALUES , *VECTOR algebra , *MATHEMATICAL analysis - Abstract
Abstract: Let be the skew adjacency matrix of the oriented graph of order n and be all eigenvalues of . The skew spectral radius of is defined as . In this paper, we investigate oriented graphs whose skew spectral radii do not exceed 2. [Copyright &y& Elsevier]
- Published
- 2013
- Full Text
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29. Epidemic phase transition of the SIS type in networks
- Author
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P. Van Mieghem
- Subjects
Physics ,Phase transition ,Coupling strength ,Mathematical analysis ,General Physics and Astronomy ,Derivative ,Type (model theory) ,01 natural sciences ,010305 fluids & plasmas ,0103 physical sciences ,Quantitative Biology::Populations and Evolution ,Fraction (mathematics) ,Adjacency matrix ,010306 general physics ,Eigenvalues and eigenvectors - Abstract
By making only one approximation of a mean-eld type, we determine the nature of the SIS type of epidemic phase transition in any network: the steady-state fraction of infected nodes y∞ is linear in (� 1 c � 1 ) for effective infection rates �↓�c, the derivative of y∞ at the epidemic thresholdc= 1 �1 is exactly computed and depends on the largest eigenvalue �1 of the adjacency matrix and on the rst- and third-order moments of the corresponding eigenvector. Since coupled oscillators in a network synchonize at a coupling strength proportional to 1 �1 ,a similar characterization of the phase transition is suggested. The behavior of y∞ aroundc was the missing part in the general steady-state theory of a SIS-type epidemic ona network. Copyright c �EPLA, 2012
- Published
- 2012
30. Bicyclic graphs with small positive index of inertia
- Author
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Yu, Guihai, Feng, Lihua, and Wang, Qingwen
- Subjects
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GRAPH theory , *EIGENVALUES , *MATRICES (Mathematics) , *NUMBER theory , *MATHEMATICAL analysis , *NUMERICAL analysis - Abstract
Abstract: The positive index of inertia of a graph G, denoted by , is the number of the positive eigenvalues of the adjacency matrix of G. In this paper, we investigate the minimal positive index of inertia among all bicyclic graphs of order n with pendant vertices, and characterize the bicyclic graphs with positive index 1 or 2 among all bicyclic graphs of order n. [Copyright &y& Elsevier]
- Published
- 2013
- Full Text
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31. Proof of conjectures on adjacency eigenvalues of graphs
- Author
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Das, Kinkar Ch.
- Subjects
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PROOF theory , *LOGICAL prediction , *EIGENVALUES , *GRAPH theory , *DIRECTED graphs , *LINEAR algebra , *MATHEMATICAL analysis - Abstract
Abstract: Let be a simple graph of order with triangle(s). Also let be the eigenvalues of the adjacency matrix of graph . X. Yong [X. Yong, On the distribution of eigenvalues of a simple undirected graph, Linear Algebra Appl. 295 (1999) 73–80] conjectured that (i) is complete if and only if and also (ii) is complete if and only if . Here we disprove this conjecture by a counter example. Wang et al. [J.F. Wang, F. Belardo, Q.X. Huang, B. Borovićanin, On the two largest Q-eigenvalues of graphs, Discrete Math. 310 (2010) 2858–2866] conjectured that friendship graph is determined by its adjacency spectrum. Here we prove this conjecture. The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity of a graph is the mean value of eccentricities of all vertices of . Moreover, we mention three conjectures, obtained by the system AutoGraphiX, about the average eccentricity , girth and the spectral radius of graphs (see Aouchiche (2006) [1], available online at http://www.gerad.ca/~agx/). We give a proof of one conjecture and disprove two conjectures by counter examples. [Copyright &y& Elsevier]
- Published
- 2013
- Full Text
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32. Maximum order of trees and bipartite graphs with a given rank
- Author
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Ghorbani, E., Mohammadian, A., and Tayfeh-Rezaie, B.
- Subjects
- *
TREE graphs , *BIPARTITE graphs , *GRAPH theory , *PATHS & cycles in graph theory , *MATHEMATICAL analysis , *COMBINATORICS - Abstract
Abstract: The rank of a graph is that of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. We determine the maximum order of reduced trees as well as bipartite graphs with a given rank and characterize those graphs achieving the maximum order. [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
- View/download PDF
33. The Spectrum of Magnetic Schrödinger Operators on a Graph with Periodic Structure
- Author
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Tomoyuki Shirai and Yusuke Higuchi
- Subjects
discrete magnetic Laplacian ,discrete spectral geometry ,Spectral graph theory ,twisted operator ,Mathematical analysis ,Quartic graph ,Physics::Fluid Dynamics ,Graph energy ,phase transition ,Covering graph ,Integral graph ,abelian covering graph ,Adjacency matrix ,Periodic graph (geometry) ,Laplacian matrix ,Analysis ,Mathematics - Abstract
For discrete magnetic Schrodinger operators on covering graphs of a finite graph, we investigate two spectral properties: (1) the relationship between the spectrum of the operator on the covering graph and that on a finite graph, (2) the analyticity of the bottom of the spectrum with respect to magnetic flow. Also we compute the second derivative of the bottom of the spectrum and represent it in terms of geometry of a graph.
- Published
- 1999
34. Determinants of box products of paths
- Author
-
Pragel, Daniel
- Subjects
- *
GRAPH theory , *PATHS & cycles in graph theory , *LENGTH measurement , *INTEGERS , *MATRICES (Mathematics) , *MATHEMATICAL analysis - Abstract
Abstract: Suppose that is the graph obtained by taking the box product of a path of length and a path of length . Let be the adjacency matrix of . In 1996, Rara showed that, if , then . We extend this result to allow and to be any positive integers, and show that [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
- View/download PDF
35. The Jordan canonical form for a class of zero–one matrices
- Author
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Cardon, David A. and Tuckfield, Bradford
- Subjects
- *
MATRICES (Mathematics) , *DIRECTED graphs , *MATHEMATICAL forms , *MATHEMATICAL functions , *GEOMETRIC connections , *MATHEMATICAL analysis , *GRAPH theory - Abstract
Abstract: Let be a function. Let be the matrix defined by if for some i and j and otherwise. We describe the Jordan canonical form of the matrix in terms of the directed graph for which is the adjacency matrix. We discuss several examples including a connection with the Collatz conjecture. [Copyright &y& Elsevier]
- Published
- 2011
- Full Text
- View/download PDF
36. On the nullity of graphs with pendant trees
- Author
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Gong, Shi-Cai, Fan, Yi-Zheng, and Yin, Zhi-Xiang
- Subjects
- *
GRAPH theory , *TREE graphs , *MATRICES (Mathematics) , *EIGENVALUES , *SET theory , *MATHEMATICAL analysis - Abstract
Abstract: The nullity of a graph is defined as the multiplicity of the eigenvalue zero in the spectrum of the adjacency matrix of the graph. We investigate a class of graphs with pendant trees, and express the nullity of such graph in terms of that of its subgraphs. As an application of our results, we characterize unicyclic graphs with a given nullity. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
37. Interlacing eigenvalues on some operations of graphs
- Author
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Wu, Bao-Feng, Shao, Jia-Yu, and Liu, Yue
- Subjects
- *
EIGENVALUES , *CHARTS, diagrams, etc. , *MATRICES (Mathematics) , *LAPLACIAN operator , *SPECTRUM analysis , *MATHEMATICAL analysis - Abstract
Abstract: In this paper, we focus on some operations of graphs and give a kind of eigenvalue interlacing in terms of the adjacency matrix, standard Laplacian, and normalized Laplacian. Also, we explore some applications of this interlacing. [Copyright &y& Elsevier]
- Published
- 2009
- Full Text
- View/download PDF
38. Minimizing the least eigenvalues of unicyclic graphs with application to spectral spread
- Author
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Fan, Yi-Zheng, Wang, Yi, and Gao, Yu-Bin
- Subjects
- *
MATHEMATICAL analysis , *MATRICES (Mathematics) , *EIGENVALUES , *HILL determinant - Abstract
Abstract: The spread of a graph is defined to be the difference between the largest eigenvalue and the least eigenvalue of the adjacency matrix of the graph. Let denote the set of connected unicyclic graphs of order n and girth k, and let denote the set of connected unicyclic graphs of order n. In this paper, we determine the unique graph with minimum least eigenvalue (respectively, the unique graph with maximum spread) among all graphs in . We, finally, characterize the unique graph with minimum least eigenvalue (respectively, the unique graph with maximum spread) among all graphs in . [Copyright &y& Elsevier]
- Published
- 2008
- Full Text
- View/download PDF
39. Some relations between rank of a graph and its complement
- Author
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Akbari, Saieed, Alipour, Alireza, Boroojeni, Javad Ebrahimi, Ghorbani, Ebrahim, and Shirazi, Mirhamed Mirjalalieh
- Subjects
- *
GRAPH theory , *COMBINATORICS , *LINEAR algebra , *MATHEMATICAL analysis - Abstract
Abstract: Let G be a graph of order n and rank(G) denotes the rank of its adjacency matrix. Clearly, . In this paper we characterize all graphs G such that or n +2. Also for every integer n ⩾5 and any k, 0⩽ k ⩽ n, we construct a graph G of order n, such that . [Copyright &y& Elsevier]
- Published
- 2007
- Full Text
- View/download PDF
40. The largest eigenvalue of unicyclic graphs
- Author
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Hu, Shengbiao
- Subjects
- *
MATRICES (Mathematics) , *MATHEMATICS , *MATHEMATICAL analysis , *GRAPH theory - Abstract
Abstract: Let G be a simple graph. Let and denote the largest eigenvalue of the adjacency matrix and the Laplacian matrix of G, respectively. Let denote the largest vertex degree. If G has just one cycle, thenThe equality holds if and only if . AndThe equality holds if and only if , n is even. [Copyright &y& Elsevier]
- Published
- 2007
- Full Text
- View/download PDF
41. 3-Regular digraphs with optimum skew energy
- Author
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Gong, Shi-Cai and Xu, Guang-Hui
- Subjects
- *
DIRECTED graphs , *MATHEMATICAL singularities , *RANDOM walks , *COMBINATORICS , *GRAPH connectivity , *MATHEMATICAL analysis - Abstract
Abstract: The skew energy of a digraph D is defined as the sum of the singular values of its skew adjacency matrix . In this paper, we first interpret the entries of the power of the skew adjacency matrix of a digraph in terms of the number of its walks and then focus on the question posed by Adiga et al. [C. Adiga, R. Balakrishnan, Wasin So, The skew energy of a graph, Linear Algebra Appl. 432 (2010) 1825–1835] of determining all -regular connected digraphs with optimum skew energy. [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
- View/download PDF
42. CONSTRUCTION OF COSPECTRAL REGULAR GRAPHS.
- Author
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Bapat, Ravindra B. and Karimi, Masoud
- Subjects
- *
REGULAR graphs , *POLYNOMIALS , *EIGENVALUES , *MULTIPLICITY (Mathematics) , *INTEGRALS , *MATHEMATICAL analysis - Abstract
Graphs G and H are called cospectral if they have the same characteristic polynomial, equivalently, if they have the same eigenvalues considering multiplicities. In this article we introduce a construction to produce pairs of cospectral regular graphs. We also investigate conditions under which the graphs are integral. [ABSTRACT FROM AUTHOR]
- Published
- 2016
43. Cospectrality of graphs.
- Author
-
Abdollahi, Alireza and Oboudi, Mohammad Reza
- Subjects
- *
GRAPH theory , *ISOMORPHISM (Mathematics) , *EUCLIDEAN algorithm , *MATHEMATICAL analysis , *GEOMETRY - Abstract
Abstract: Richard Brualdi proposed in Stevanivić (2007) [6] the following problem: (Problem AWGS.4) Let and be two nonisomorphic graphs on n vertices with spectra respectively. Define the distance between the spectra of and as Define the cospectrality of by Let In this paper we study Problem A and determine the cospectrality of certain graphs by the Euclidian distance. Let denote the complete graph on n vertices, denote the null graph on n vertices and denote the disjoint union of the with isolated vertices, where . In this paper we find , , ( ) and . [Copyright &y& Elsevier]
- Published
- 2014
- Full Text
- View/download PDF
44. New upper bounds on the spectral radius of trees with the given number of vertices and maximum degree.
- Author
-
Song, Haizhou, Wang, Qiufen, and Tian, Lulu
- Subjects
- *
TREE graphs , *NUMBER theory , *ESTIMATION theory , *MATHEMATICAL inequalities , *MATHEMATICAL analysis - Abstract
Abstract: This paper studies the problem of estimating the spectral radius of trees with the given number of vertices and maximum degree. We obtain the new upper bounds on the spectral radius of the trees, and the results are the best upper bounds expressed by the number of vertices and maximum degree, at present. Let be a tree on n vertices with maximum degree Δ, where . Denote by the spectral radius of T. We prove that [(1)] if , then , and equality holds if and only if T is an almost completely full-degree tree of 3 levels; [(2)] if , then , and equality holds if and only if T is a completely full-degree tree of 3 levels; [(3)] if , then , where . [Copyright &y& Elsevier]
- Published
- 2013
- Full Text
- View/download PDF
45. Regular bipartite graphs with three distinct non-negative eigenvalues
- Author
-
Koledin, Tamara and Stanić, Zoran
- Subjects
- *
BIPARTITE graphs , *GRAPH theory , *EIGENVALUES , *INCIDENCE algebras , *SPECTRAL geometry , *MATHEMATICAL analysis - Abstract
Abstract: We derive some structural and spectral properties of regular bipartite graphs with three distinct non-negative eigenvalues. Next, we consider the relations between these graphs and two-class partially balanced incomplete block designs, and we present a number of situations when the graphs we consider are in fact the incidence graphs of those designs. As a consequence, we give a several constructions of connected regular bipartite graphs with six distinct eigenvalues, and we also determine all such graphs with degree 3, and all such graphs on at most 20 vertices. [Copyright &y& Elsevier]
- Published
- 2013
- Full Text
- View/download PDF
46. On least eigenvalues and least eigenvectors of real symmetric matrices and graphs
- Author
-
Xing, Rundan and Zhou, Bo
- Subjects
- *
EIGENVALUES , *EIGENVECTORS , *SYMMETRIC matrices , *GRAPH theory , *MATHEMATICAL bounds , *MATHEMATICAL analysis - Abstract
Abstract: We give an upper bound for the least eigenvalue of a principal submatrix of a real symmetric matrix with zero diagonal, from which we establish an upper bound for the least eigenvalue of a graph when some vertices are removed using the components of the least eigenvector(s). We give lower and upper bounds for the least eigenvalue of a graph when some edges are removed. We also establish bounds for the components of the least eigenvector(s) of a real symmetric matrix and a graph. [Copyright &y& Elsevier]
- Published
- 2013
- Full Text
- View/download PDF
47. An oriented hypergraphic approach to algebraic graph theory
- Author
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Reff, Nathan and Rusnak, Lucas J.
- Subjects
- *
HYPERGRAPHS , *GRAPH theory , *PATHS & cycles in graph theory , *LAPLACIAN matrices , *COMBINATORICS , *MATHEMATICAL analysis - Abstract
Abstract: An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of or . We define the adjacency, incidence and Laplacian matrices of an oriented hypergraph and study each of them. We extend several matrix results known for graphs and signed graphs to oriented hypergraphs. New matrix results that are not direct generalizations are also presented. Finally, we study a new family of matrices that contains walk information. [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
- View/download PDF
48. Some graphs whose second largest eigenvalue does not exceed
- Author
-
Stanić, Zoran
- Subjects
- *
TREE graphs , *GRAPH theory , *EIGENVALUES , *BIPARTITE graphs , *PATHS & cycles in graph theory , *MATHEMATICAL analysis - Abstract
Abstract: We determine all trees whose second largest eigenvalue does not exceed . Next, we consider two classes of bipartite graphs, regular and semiregular, with small number of distinct eigenvalues. For all graphs considered we determine those whose second largest eigenvalue is equal to . Some additional results are also given. [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
- View/download PDF
49. The spectrum and the signless Laplacian spectrum of coronae
- Author
-
Cui, Shu-Yu and Tian, Gui-Xian
- Subjects
- *
SPECTRAL theory , *HARMONIC functions , *GRAPH connectivity , *GRAPH theory , *ISOMORPHISM (Mathematics) , *MATHEMATICAL analysis - Abstract
Abstract: Let be two simple connected graphs. Denote the corona and the edge corona of by and , respectively. In this paper, we first introduce a new invariant, the -coronal of a graph matrix , where the matrix is associated with a graph in a prescribed way. Then it is used to compute the signless Laplacian spectrum of and in terms of the signless Laplacian spectrum of and . In addition, the spectrum of are also given in terms of the spectrum of and . Finally, as an application of these results, we construct many pairs of nonisomorphic signless Laplacian cospectral graphs. [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
- View/download PDF
50. The spectral characterization of -graphs
- Author
-
Liu, Fenjin, Huang, Qiongxiang, Wang, Jianfeng, and Liu, Qinghai
- Subjects
- *
SPECTRAL theory , *GRAPH theory , *PATHS & cycles in graph theory , *COMBINATORICS , *MATHEMATICAL analysis , *LINEAR algebra - Abstract
Abstract: A -graph is a graph consisting of two cycles and with just a vertex in common. Fan and Luo in Theorem 4.4 alleged that a -graph is determined by its adjacency spectrum if it contains no cycle . However, according to Theorem 6.3 we find that the result in Theorem 4.4 is not completely correct. In this paper, we prove that is DAS if and only if , and has a unique cospectral mate if . [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
- View/download PDF
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