Maximum Laplacian energy among threshold graphs

Laplacian energy of partial complement of a graph

On the normalized spectrum of threshold graphs

Paired threshold graphs

Let $A(G)$ and $D(G)$ be the adjacency matrix and the degree diagonal matrix of a graph $G$, respectively. Then $L(G)=D(G)-A(G)$ is called Laplacian matrix of the graph $G$. Let $G$ be a graph with $n$ vertices and $m$ edges. Then the $LI$-matrix of $G$ is defined as $LI(G)=L(G)-\frac{2m}{n}I_n$, where $I_n$ is the identity matrix. In this paper, we are interested in extremal properties of the Ky Fan $k$-norm of the $LI$-matrix of graphs, which is closely related to the well known problems and results in spectral graph theory, such as the Laplacian spectral radius, the Laplacian spread, the sum of the $k$ largest Laplacian eigenvalues, the Laplacian energy, and other parameters. Some bounds on the Ky Fan $k$-norm of the $LI$-matrix of graphs are given, and the extremal graphs are partly characterized. In addition, upper and lower bounds on the Ky Fan $k$-norm of $LI$-matrix of trees, unicyclic graphs, and bicyclic graphs are determined, and the corresponding extremal graphs are characterized.

Fault diagnosis of roller bearings based on Laplacian energy feature extraction of path graphs

The threshold model can be used to generate random networks of arbitrary size with given local properties such as the degree distribution, clustering, and degree correlation. We summarize the properties of networks created using the threshold model and present an alternative deterministic construction. These networks are threshold graphs and therefore contain a highly compressible layered structure and allow computation of important network properties in linear time. We show how to construct arbitrarily large, sparse, threshold networks with (approximately) any prescribed degree distribution or Laplacian spectrum. Control of the spectrum allows careful study of the synchronization properties of threshold networks including the relationship between heterogeneous degrees and resistance to synchrony.

A concept of germ graphs and the M-function formalism are employed to construct large families of isospectral and isoscattering graphs. This approach represents a complete departure from the original approach pioneered by Sunada, where isospectral graphs are obtained as quotients of a certain large symmetric graph. Using the M-function formalism and the symmetries of the graph itself we construct isospectral and isoscattering pairs. In our approach isospectral pairs do not need to be embedded into a larger symmetric graph as in Sunada's approach. We demonstrate that the introduced formalism can also be extended to graphs with dissipation. The theoretical predictions are validated experimentally using microwave networks emulating open quantum graphs with dissipation.

The energy analysis of the newly introduced n-wheel graph, employs diverse matrix representations such as the adjacency matrix, Laplacian matrix, and maximum degree matrix. This novel graph model resembles a hierarchical wireless sensor network (WSN), with a central hub serving as the communication center. The graph is organized into cycles, reflecting tiers of devices or sensors, with the hub managing wireless communication across these tiers. Through comparative analysis of energy variations, particularly focusing on ordinary energy, Laplacian energy, and maximum degree energy, offers a deeper understanding on the potential benefits of the n-wheel graph model, guiding future research and practical applications in the design of advanced hierarchical network structures.

This paper considers commuting graphs over the semidihedral group SD8n. We compute their eigenvalues and obtain that these commuting graphs are not hyperenergetic for odd n≥15 or even n≥2. We further compute the Laplacian spectrum, the Laplacian energy and the number of spanning trees of the commuting graphs over SD8n. We also discuss vertex connectivity, planarity, and minimum disconnecting sets of these graphs and prove that these commuting graphs are not Hamiltonian.

The Laplacian energy of a graph refers to the total absolute differences between its eigenvalues and the graph’s mean degree. These eigenvalues are derived from the Laplacian matrix, L which defined as a square matrix with entries Lii = di (the vertex degree of vi ), Lij=-1 if two distinct vertices are adjacent and otherwise Lij = 0. The graph is said to be the zero divisor graph of a commutative ring R is when all elements of nonzero zero divisors of R be its vertices and two distinct vertices are adjacent if and only if they are commute and equal to zero. This paper constructs the zero divisor graphs of Z3a where a=5,7,11,13 and 17 by using Python software. Subsequently, the general formula for the Laplacian spectrum and Laplacian energy are derived from the constructed graph, with an example provided to illustrate the main theorems.

We discuss scattering from pairs of isospectral quantum graphs constructed using the method described in the papers of Parzanchevski, Band and Ben-Shach.It was shown in the paper of Band et al. that scattering matrices of such graphs have the same spectrum and polar structure, provided that innite leads are attached in a way which preserves the symmetry of isospectral construction.In the current paper we compare this result with the conjecture put forward by Okada et al. that the pole distribution of scattering matrices in the exterior of isospectral domains in R 2 is dierent.

This study is concerned with the minimal controllability problem of a connected threshold graph following the Laplacian dynamics. The goal is to find the minimum number of controllers and a small set of vertices for the controllers to connect to render the graph Laplacian controllable. A simple algorithm is provided to generate a spanning set of orthogonal Laplacian eigenvectors of the graph from a straightforward computation on its Laplacian matrix. A necessary and sufficient condition for the graph to be Laplacian controllable is then proposed. The condition suggests that the minimum number of controllers to make a connected threshold graph Laplacian controllable is the maximum multiplicity of Laplacian eigenvalues of the graph, and this minimum can be achieved using a binary control matrix. If a controller can be connected to one vertex only, the minimum number is the difference between the number of vertices in the graph and the number of vertices with different degrees. The condition also implies that the controllers ensuring the Laplacian controllability should be connected to the vertices with repeating degrees to break the symmetry of the network topology. Several examples are provided to illustrate the authors' results.

Integer-Distance Graphs

Given a special kind of symmetric matrix, we present a decomposition technique that preserves its spectrum. We transform this result into an algorithm for graphs. The algorithm disconnects the graph, resulting in smaller matrices, reducing the complexity of computing the graph spectrum. This technique may be seen as a unified approach of several decomposition techniques present in the literature for different instances of matrices and classes of graphs. As an application, we use the algorithm for three classes of graphs: threshold graphs, generalized Bethe trees, and multifan graphs, obtaining expressions for their spectra, for various matrices.

Graph products and their structural properties have been studied extensively by many researchers. We investigate the Laplacian eigenvalues and eigenvectors of the product graphs for the four standard products, namely, the Cartesian product, the direct product, the strong product and the lexicographic product. A complete characterization of Laplacian spectrum of the Cartesian product of two graphs has been done by Merris. We give an explicit complete characterization of the Laplacian spectrum of the lexicographic product of two graphs using the Laplacian spectra of the factors. For the other two products, we describe the complete spectrum of the product graphs in some particular cases. We supply some new results relating to the algebraic connectivity of the product graphs. We describe the characteristic sets for the Cartesian product and for the lexicographic product of two graphs. As an application we construct new classes of Laplacian integral graphs.