407 research outputs found
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 HĂŒckel 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
What is the meaning of the graph energy after all?
For a simple graph with eigenvalues of the adjacency matrix
, the energy of the graph
is defined by . Myriads of papers have been
published in the mathematical and chemistry literature about properties of this
graph invariant due to its connection with the energy of (bipartite) conjugated
molecules. However, a structural interpretation of this concept in terms of the
contributions of even and odd walks, and consequently on the contribution of
subgraphs, is not yet known. Here, we find such interpretation and prove that
the (adjacency) energy of any graph (bipartite or not) is a weighted sum of the
traces of even powers of the adjacency matrix. We then use such result to find
bounds for the energy in terms of subgraphs contributing to it. The new bounds
are studied for some specific simple graphs, such as cycles and fullerenes. We
observe that including contributions from subgraphs of sizes not bigger than 6
improves some of the best known bounds for the energy, and more importantly
gives insights about the contributions of specific subgraphs to the energy of
these graphs
Maximum Estrada Index of Bicyclic Graphs
Let be a simple graph of order , let
be the eigenvalues of the
adjacency matrix of . The Esrada index of is defined as
. In this paper we determine the unique
graph with maximum Estrada index among bicyclic graphs with fixed order
Gaussianization of the spectra of graphs and networks : theory and applications
Matrix functions of the adjacency matrix are very useful for understanding important structural properties of graphs and networks, such as communicability, node centrality, bipartivity and many more. Here we propose a new matrix function based on the Gaussianization of the adjacency matrix of a graph. This function gives more weight to a selected reference eigenvalue λref, which may be located in any region of the graph spectra.
In particular, we study the Gaussian Estrada indices for two reference eigenvalues 0 and -1 separately. In each case, we obtain bounds for this index in simple graphs. We also obtain formulas for the Gaussian Estrada index of Erdos-Renyi random graphs as well as for the Barabasi-Albert graphs. Moreover, for λref = 0, we show that in real-world networks this index is related to the existence of important structural patterns, such as complete bipartite subgraphs (bicliques). Such bicliques appear naturally in many real-world networks as a consequence of evolutionary processes giving rise to them.
In addition, we fold the graph spectrum at a given pair of reference eigenvalues, then exponentiate the resulting folded graph spectrum to produce the double Gaussian function of the graph adjacency matrix which give more importance to the reference eigenvalues than to the rest of the spectrum. Based on evidence from mathematical chemistry we focus here our attention on the reference eigenvalues ±1. They enclose most of the HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) of organic molecular graphs.
We prove several results for the trace of the double Gaussian adjacency matrix of simple graphs - the double Gaussian Estrada index - and we apply this index to the classification of polycyclic aromatic hydrocarbons (PAHs) as carcinogenic or non-carcinogenic. We discover that local indices based on the previously developed matrix function allow to classify correctly 100% of the PAHs analyzed. In general, folding the spectrum of the adjacency matrix of networks characterizes important structural information not described in previously used matrix functions of graphs.Matrix functions of the adjacency matrix are very useful for understanding important structural properties of graphs and networks, such as communicability, node centrality, bipartivity and many more. Here we propose a new matrix function based on the Gaussianization of the adjacency matrix of a graph. This function gives more weight to a selected reference eigenvalue λref, which may be located in any region of the graph spectra.
In particular, we study the Gaussian Estrada indices for two reference eigenvalues 0 and -1 separately. In each case, we obtain bounds for this index in simple graphs. We also obtain formulas for the Gaussian Estrada index of Erdos-Renyi random graphs as well as for the Barabasi-Albert graphs. Moreover, for λref = 0, we show that in real-world networks this index is related to the existence of important structural patterns, such as complete bipartite subgraphs (bicliques). Such bicliques appear naturally in many real-world networks as a consequence of evolutionary processes giving rise to them.
In addition, we fold the graph spectrum at a given pair of reference eigenvalues, then exponentiate the resulting folded graph spectrum to produce the double Gaussian function of the graph adjacency matrix which give more importance to the reference eigenvalues than to the rest of the spectrum. Based on evidence from mathematical chemistry we focus here our attention on the reference eigenvalues ±1. They enclose most of the HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) of organic molecular graphs.
We prove several results for the trace of the double Gaussian adjacency matrix of simple graphs - the double Gaussian Estrada index - and we apply this index to the classification of polycyclic aromatic hydrocarbons (PAHs) as carcinogenic or non-carcinogenic. We discover that local indices based on the previously developed matrix function allow to classify correctly 100% of the PAHs analyzed. In general, folding the spectrum of the adjacency matrix of networks characterizes important structural information not described in previously used matrix functions of graphs
Singlet Oxygen Generation by Porphyrins and Metalloporphyrins Revisited: a Quantitative Structure-property Relationship (QSPR) Study
state followed by formation of singlet oxygen (1O2), which is a highly reactive species and mediates various oxidative processes. The design of advanced sensitizers based on porphyrin compounds have attracted significant attention in recent years. However, it is still difficult to predict the efficiency of singlet oxygen generation for a given structure. Our goal was to develop a quantitative structure-property relationship (QSPR) model for the fast virtual screening and prediction of singlet oxygen quantum yields for pophyrins and metalloporphyrins. We performed QSPR analysis of a dataset containing 32 compounds, including various porphyrins and their analogues (chlorins and bacteriochlorins). Quantum-chemical descriptors were calculated using Density Functional Theory (DFT), namely B3LYP and M062X functionals. Three different machine learning methods were used to develop QSPR models: random forest regression (RFR), support vector regression (SVR), and multiple linear regression (MLR). The optimal QSPR model «structure â singlet oxygen generation quantum yield» obtained using RFR method demonstrated high determination coefficient for the training set (R2âŻ=âŻ0.949) and the highest predicting ability for the test set (pred_R2 = 0.875). This proves that the developed QSPR method is realiable and can be directly applied in the studies of singlet oxygen generation both for free base porphyrins and their metal complexes. We believe that QSPR approach developed in this study can be useful for the search of new poprhyrin photosensitizers with enhanced singlet oxygen generation ability
Isospectral and Subspectral Molecules
Isospectral molecules are non-identical structures which possess
the same spectrum of eigenvalues. Methods for recognizing
isospectrality, procedures of Heilbronner, Herndon and .Zivkovic for
constructing new isospectral mates, and the specification of the
relationship among the eigenvectors of the adjacency matrix of
isospectral pairs are discussed here.
Instances of isospectral graphs are relatively rare. There are
many cases, however, in which the spectrum of one molecular graph
contains the spectrum of a second, smaller graph. In such cases, the
larger, composite, graph and the smaller, component graph are said
to be subspectral. Methods of McClelland, Hall and D\u27Amato for
determining subspectrality of graphs are reviewed in detail. It
appears that all known cases of subspectral molecules, but one, can
be explained by various decomposition or factorization schemes.
No chemical evidence is found so far that shows a relationship
among the measured properties of isospectral or subspectral molecules.
However, the existence of isospectral and subspectral molecules
prevented the use of characteristic polynomial for the unique
characterization of molecules in various classification schemes and
in computerized chemical documentation
Exploring Complex Networks with Graph Investigator Research Application
This paper describes Graph Investigator, the application intended for analysis of complex networks. A rich set of application functions is briefly described including graph feature generation, comparison, visualization and edition. The program enables to analyze global and local structural properties of networks with the use of various descriptors derived from graph theory. Furthermore, it allows to quantify inter-graph similarity by embedding graph patterns into low-dimensional space or distance measurement based on feature vectors. The set of available graph descriptors includes over eighty statistical and algebraic measures. We present two examples of real-world networks analysis performed with Graph Investigator: comparison of brain vasculature with structurally similar artificial networks and analysis of vertices importance in a macaque cortical connectivity network. The third example describes tracking parameters of artificial vascular network evolving in the process of angiogenesis, modelled with the use of cellular automata
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