Related papers: Exploring the "Middle Earth" of Network Spectra vi…
Let $G$ be a graph on $n$ vertices and $\lambda_1,\lambda_2,\ldots,\lambda_n$ its eigenvalues. The Estrada index of $G$ is an invariant that is calculated from the eigenvalues of the adjacency matrix of a graph. In this paper, we present…
Many natural and social systems develop complex networks, that are usually modelled as random graphs. The eigenvalue spectrum of these graphs provides information about their structural properties. While the semi-circle law is known to…
Let $G$ be a simple graph of order $n$, let $\lambda_1(G),\lambda_2(G),...,\lambda_n(G)$ be the eigenvalues of the adjacency matrix of $G$. The Esrada index of $G$ is defined as $EE(G)=\sum_{i=1}^{n}e^{\lambda_i(G)}$. In this paper we…
The spectrum of a network or graph $G=(V,E)$ with adjacency matrix $A$, consists of the eigenvalues of the normalized Laplacian $L= I - D^{-1/2} A D^{-1/2}$. This set of eigenvalues encapsulates many aspects of the structure of the graph,…
We apply random matrix theory to complex networks. We show that nearest neighbor spacing distribution of the eigenvalues of the adjacency matrices of various model networks, namely scale-free, small-world and random networks follow…
Matrix functions play an important role in applied mathematics. In network analysis, in particular, the exponential of the adjacency matrix associated with a network provides valuable information about connectivity, as well as about the…
The D-eigenvalues {\mu}_1,{\mu}_2,...,{\mu}_{n} of a connected graph G are the eigenvalues of its distance matrix. The distance Estrada index of G is defined in [15] as DEE=DEE(G)=\Sigma_{i=1}^n e^{{\mu}_{i}} In this paper, we give better…
The spectrum of the nonbacktracking matrix associated to a network is known to contain fundamental information regarding percolation properties of the network. Indeed, the inverse of its leading eigenvalue is often used as an estimate for…
The Estrada index of a graph $G$ is defined as $EE(G)=\sum_{i=1}^ne^{\lambda_i}$, where $\lambda_1,$ $ \lambda_2,\ldots, \lambda_n$ are the eigenvalues of the adjacency matrix of $G$. In this paper, we characterize the unique bipartite…
Networks are often studied using the eigenvalues of their adjacency matrix, a powerful mathematical tool with a wide range of applications. Since in real systems the exact graph structure is not known, researchers resort to random graphs to…
We introduce a new matrix function for studying graphs and real-world networks based on a double-factorial penalization of walks between nodes in a graph. This new matrix function is based on the matrix error function. We find a very good…
Many complex systems can be represented as networks, and how a network breaks up into subnetworks or communities is of wide interest. However, the development of a method to detect nodes important to communities that is both fast and…
Determining the effect of structural perturbations on the eigenvalue spectra of networks is an important problem because the spectra characterize not only their topological structures, but also their dynamical behavior, such as…
Typically, graph structures are represented by one of three different matrices: the adjacency matrix, the unnormalised and the normalised graph Laplacian matrices. The spectral (eigenvalue) properties of these different matrices are…
We study the structural characteristics of complex networks using the representative eigenvectors of the adjacent matrix. The probability distribution function of the components of the representative eigenvectors are proposed to describe…
Quantifying the eigenvalue spectra of large random matrices allows one to understand the factors that contribute to the stability of dynamical systems with many interacting components. This work explores the effect that the interaction…
Let $\lambda_1,\lambda_2,\cdots,\lambda_n$ be the eigenvalues of the distance matrix of a connected graph $G$. The distance Estrada index of $G$ is defined as $DEE(G)=\sum_{i=1}^ne^{\lambda_i}$. In this note, we present new lower and upper…
Let $G$ be a graph on $n$ vertices and $\lambda_1,\lambda_2,\ldots,\lambda_n$ its eigenvalues. The Estrada index of $G$ is defined as $EE(G)=\sum_{i=1}^n e^{\lambda_i}.$ In this work, using a different demonstration technique, new lower…
Centrality measures are used in network science to identify the most important vertices for transmission of information and dynamics on a graph. One of these measures, introduced by Estrada and collaborators, is the $\beta$-subgraph…
The Laplacian matrix of a simple graph is the difference of the diagonal matrix of vertex degree and the (0,1) adjacency matrix. In the past decades, the Laplacian spectrum has received much more and more attention, since it has been…