380 research outputs found
New Bounds for the Harmonic Energy and Harmonic Estrada index of Graphs
Let be a finite simple undirected graph with vertices and edges. The Harmonic energy of a graph , denoted by , is defined as the sum of the absolute
values of all Harmonic eigenvalues of . The Harmonic Estrada index of a graph , denoted by , is defined as
where are the - of .
In this paper we present some new bounds for and in terms of number of vertices, number of edges and the sum-connectivity index
The Physics of Communicability in Complex Networks
A fundamental problem in the study of complex networks is to provide
quantitative measures of correlation and information flow between different
parts of a system. To this end, several notions of communicability have been
introduced and applied to a wide variety of real-world networks in recent
years. Several such communicability functions are reviewed in this paper. It is
emphasized that communication and correlation in networks can take place
through many more routes than the shortest paths, a fact that may not have been
sufficiently appreciated in previously proposed correlation measures. In
contrast to these, the communicability measures reviewed in this paper are
defined by taking into account all possible routes between two nodes, assigning
smaller weights to longer ones. This point of view naturally leads to the
definition of communicability in terms of matrix functions, such as the
exponential, resolvent, and hyperbolic functions, in which the matrix argument
is either the adjacency matrix or the graph Laplacian associated with the
network. Considerable insight on communicability can be gained by modeling a
network as a system of oscillators and deriving physical interpretations, both
classical and quantum-mechanical, of various communicability functions.
Applications of communicability measures to the analysis of complex systems are
illustrated on a variety of biological, physical and social networks. The last
part of the paper is devoted to a review of the notion of locality in complex
networks and to computational aspects that by exploiting sparsity can greatly
reduce the computational efforts for the calculation of communicability
functions for large networks.Comment: Review Article. 90 pages, 14 figures. Contents: Introduction;
Communicability in Networks; Physical Analogies; Comparing Communicability
Functions; Communicability and the Analysis of Networks; Communicability and
Localization in Complex Networks; Computability of Communicability Functions;
Conclusions and Prespective
Spectral graph theory : from practice to theory
Graph theory is the area of mathematics that studies networks, or graphs. It arose from the need to analyse many diverse network-like structures like road networks, molecules, the Internet, social networks and electrical networks. In spectral graph theory, which is a branch of graph theory, matrices are constructed from such graphs and analysed from the point of view of their so-called eigenvalues and eigenvectors. The first practical need for studying graph eigenvalues was in quantum chemistry in the thirties, forties and fifties, specifically to describe the Hückel molecular orbital theory for unsaturated conjugated hydrocarbons. This study led to the field which nowadays is called chemical graph theory. A few years later, during the late fifties and sixties, graph eigenvalues also proved to be important in physics, particularly in the solution of the membrane vibration problem via the discrete approximation of the membrane as a graph. This paper delves into the journey of how the practical needs of quantum chemistry and vibrating membranes compelled the creation of the more abstract spectral graph theory. Important, yet basic, mathematical results stemming from spectral graph theory shall be mentioned in this paper. Later, areas of study that make full use of these mathematical results, thus benefitting greatly from spectral graph theory, shall be described. These fields of study include the P versus NP problem in the field of computational complexity, Internet search, network centrality measures and control theory.peer-reviewe
"Clumpiness" Mixing in Complex Networks
Three measures of clumpiness of complex networks are introduced. The measures
quantify how most central nodes of a network are clumped together. The
assortativity coefficient defined in a previous study measures a similar
characteristic, but accounts only for the clumpiness of the central nodes that
are directly connected to each other. The clumpiness coefficient defined in the
present paper also takes into account the cases where central nodes are
separated by a few links. The definition is based on the node degrees and the
distances between pairs of nodes. The clumpiness coefficient together with the
assortativity coefficient can define four classes of network. Numerical
calculations demonstrate that the classification scheme successfully
categorizes 30 real-world networks into the four classes: clumped assortative,
clumped disassortative, loose assortative and loose disassortative networks.
The clumpiness coefficient also differentiates the Erdos-Renyi model from the
Barabasi-Albert model, which the assortativity coefficient could not
differentiate. In addition, the bounds of the clumpiness coefficient as well as
the relationships between the three measures of clumpiness are discussed.Comment: 47 pages, 11 figure
"Clumpiness" Mixing in Complex Networks
Three measures of clumpiness of complex networks are introduced. The measures
quantify how most central nodes of a network are clumped together. The
assortativity coefficient defined in a previous study measures a similar
characteristic, but accounts only for the clumpiness of the central nodes that
are directly connected to each other. The clumpiness coefficient defined in the
present paper also takes into account the cases where central nodes are
separated by a few links. The definition is based on the node degrees and the
distances between pairs of nodes. The clumpiness coefficient together with the
assortativity coefficient can define four classes of network. Numerical
calculations demonstrate that the classification scheme successfully
categorizes 30 real-world networks into the four classes: clumped assortative,
clumped disassortative, loose assortative and loose disassortative networks.
The clumpiness coefficient also differentiates the Erdos-Renyi model from the
Barabasi-Albert model, which the assortativity coefficient could not
differentiate. In addition, the bounds of the clumpiness coefficient as well as
the relationships between the three measures of clumpiness are discussed.Comment: 47 pages, 11 figure
On Some Aspects of the Generalized Petersen Graph
Let be a positive integer and let . The generalized Petersen graph GP(p,k) has its vertex and edge set as and . In this paper we probe its spectrum and determine the Estrada index, Laplacian Estrada index, signless Laplacian Estrada index, normalized Laplacian Estrada index, and energy of a graph. While obtaining some interesting results, we also provide relevant background and problems
Decay properties of spectral projectors with applications to electronic structure
Motivated by applications in quantum chemistry and solid state physics, we
apply general results from approximation theory and matrix analysis to the
study of the decay properties of spectral projectors associated with large and
sparse Hermitian matrices. Our theory leads to a rigorous proof of the
exponential off-diagonal decay ("nearsightedness") for the density matrix of
gapped systems at zero electronic temperature in both orthogonal and
non-orthogonal representations, thus providing a firm theoretical basis for the
possibility of linear scaling methods in electronic structure calculations for
non-metallic systems. We further discuss the case of density matrices for
metallic systems at positive electronic temperature. A few other possible
applications are also discussed.Comment: 63 pages, 13 figure
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