Spectrum Of Laplacian Research Articles

The Laplacian Spectrum, Kirchhoff Index, and the Number of Spanning Trees of the Linear Heptagonal Networks

Let H n be the linear heptagonal networks with 2 n heptagons. We study the structure properties and the eigenvalues of the linear heptagonal networks. According to the Laplacian polynomial of H n , we utilize the method of decompositions. Thus, the Laplacian spectrum of H n is created by eigenvalues of a pair of matrices: L A and L S of order numbers 5 n + 1 and 4 n + 1 n ! / r ! n − r ! , respectively. On the basis of the roots and coefficients of their characteristic polynomials of L A and L S , we get not only the explicit forms of Kirchhoff index but also the corresponding total number of spanning trees of H n .

Laplacian spectrum and Brouwer's conjecture of a graph

Laplacian spectrum and Brouwer's conjecture of a graph

Energy of Certain Classes of Graphs Determined by Their Laplacian Degree Product Adjacency Spectrum

In this study, we investigate the Laplacian degree product spectrum and corresponding energy of four families of graphs, namely, complete graphs, complete bipartite graphs, friendship graphs, and corona products of 3 and 4 cycles with a null graph.

Order Sum Graph of a Group

The concept of the order sum graph associated with a finite group based on the order of the group and order of group elements is introduced. Some of the properties and characteristics such as size, chromatic number, domination number, diameter, circumference, independence number, clique number, vertex connectivity, spectra, and Laplacian spectra of the order sum graph are determined. Characterizations of the order sum graph to be complete, perfect, etc. are also obtained.

DISTANCE LAPLACIAN SPECTRUM OF THE COMMUTING GRAPH OF FINITE CA-GROUPS

The commuting graph of a finite group $G$, $mathcal{C}(G)$, is a simple graph with vertex set $G$ in which two vertices $x$ and $y$ are adjacent if and only if $xy = yx$. The aim of this paper is to compute the distance Laplacian spectrum and the distance Laplacian energy of the commuting graph of $CA$-groups.

Some result on integrality of several matrix representation of complete r-uniform hypergraph

The eigenvalues of matrix representation hypergraph have the possibility that all of it can be an integer or otherwise. The Laplacian integral hypergraphs are those hypergraphs whose Laplacian spectrum consists entirely of integers likewise the definition of signless Laplacian integral and Seidel integral. This research focuses on the properties of the entry matrix and formulates the spectrum of the Laplacian matrix, Signless Laplacian matrix, and Seidel Matrix of complete r-uniform hypergraphs. Hence, it can determine the integrality of the representation matrix respectively. The result concludes that complete r-uniform hypergraphs are Laplacian integral, signless Laplacian integral, and Seidel integral.

Two Kinds of Laplacian Spectra and Degree Kirchhoff Index of the Weighted Corona Networks

Recently, the study related to network has aroused wide attention of the scientific community. Many problems can be usefully represented by corona graphs or networks. Meanwhile, the weight is a vital factor in characterizing some properties of real networks. In this paper, we give complete information about the signless Laplacian spectra of the weighted corona of a graph G1 and a regular graph G2 and the complete information about the normalized Laplacian spectra of the weighted corona of two regular graphs. The corresponding linearly independent eigenvectors of all these eigenvalues are also obtained. The spanning trees’ total number and the degree Kirchhoff index of the weighted corona graph are computed.

Computing the Normalized Laplacian Spectrum and Spanning Tree of the Strong Prism of Octagonal Network

Spectrum analysis and computing have expanded in popularity in recent years as a critical tool for studying and describing the structural properties of molecular graphs. Let be the strong prism of an octagonal network On. In this study, using the normalized Laplacian decomposition theorem, we determine the normalized Laplacian spectrum of which consists of the eigenvalues of matrices ℒA and ℒS of order 3n + 1. As applications of the obtained results, the explicit formulae of the degree‐Kirchhoff index and the number of spanning trees for are on the basis of the relationship between the roots and coefficients.

Consensus of double integrators over a chain with reverse edges*

Consensus in directed networks, comprising agents modelled as double integrators, is dependent on eigenvalues of the graph Laplacian matrix of the network. Second order consensus is always guaranteed if there is a directed spanning tree without any directed cycles. However, in the presence of directed cycles, consensus cannot be guaranteed. Although the necessary and sufficient condition for consensus then depends on both the real and imaginary part of the eigenvalues of the graph Laplacian, the eigenvalue with smallest real part paves the way towards a sufficient condition for consensus. This paper will first investigate how the presence of reverse edges (leading to directed cycles) affects the Laplacian spectra of a chain network, and subsequently present a sufficient condition for consensus among double integrators in the presence of two directed cycles in a chain network.

Synchronizability of Multi-Layer Dual-Center Coupled Star Networks

In the research on complex networks, synchronizability is a significant measurement of network nature. Several research studies center around the synchronizability of single-layer complex networks and few studies on the synchronizability of multi-layer networks. Firstly, this paper calculates the Laplacian spectrum of multi-layer dual-center coupled star networks and multi-layer dual-center coupled star–ring networks according to the master stability function (MSF) and obtains important indicators reflecting the synchronizability of the above two network structures. Secondly, it discusses the relationships among synchronizability and various parameters, and numerical simulations are given to illustrate the effectiveness of the theoretical results. Finally, it is found that the two sorts of networks studied in this paper are of the same synchronizability, and compared with that of a single-center network structure, the synchronizability of two dual-center structures is relatively weaker.

Consensus Indices of Two-Layered Multi-Star Networks: An Application of Laplacian Spectrum

In this article, the convergence speed and robustness of the consensus for several dual-layered star-composed multi-agent networks are studied through the method of graph spectra. The consensus-related indices, which can measure the performance of the coordination systems, refer to the algebraic connectivity of the graph and the network coherence. In particular, graph operations are introduced to construct several novel two-layered networks, the methods of graph spectra are applied to derive the network coherence for the multi-agent networks, and we find that the adherence of star topologies will make the first-order coherence of the dual-layered systems increase some constants in the sense of limit computations. In the second-order case, asymptotic properties also exist when the index is divided by the number of leaf nodes. Finally, the consensus-related indices of the duplex networks with the same number of nodes but non-isomorphic structures have been compared and simulated, and it is found that both the first-order coherence and second-order coherence of the network D are between A and B, and C has the best first-order robustness, but it has the worst robustness in the second-order case.

Spectral detection of simplicial communities via Hodge Laplacians.

While the study of graphs has been very popular, simplicial complexes are relatively new in the network science community. Despite being a source of rich information, graphs are limited to pairwise interactions. However, several real-world networks such as social networks, neuronal networks, etc., involve interactions between more than two nodes. Simplicial complexes provide a powerful mathematical framework to model such higher-order interactions. It is well known that the spectrum of the graph Laplacian is indicative of community structure, and this relation is exploited by spectral clustering algorithms. Here we propose that the spectrum of the Hodge Laplacian, a higher-order Laplacian defined on simplicial complexes, encodes simplicial communities. We formulate an algorithm to extract simplicial communities (of arbitrary dimension). We apply this algorithm to simplicial complex benchmarks and to real higher-order network data including social networks and networks extracted using language or text processing tools. However, datasets of simplicial complexes are scarce, and for the vast majority of datasets that may involve higher-order interactions, only the set of pairwise interactions are available. Hence, we use known properties of the data to infer the most likely higher-order interactions. In other words, we introduce an inference method to predict the most likely simplicial complex given the community structure of its network skeleton. This method identifies as most likely the higher-order interactions inducing simplicial communities that maximize the adjusted mutual information measured with respect to ground-truth community structure. Finally, we consider higher-order networks constructed through thresholding the edge weights of collaboration networks (encoding only pairwise interactions) and provide an example of persistent simplicial communities that are sustained over a wide range of the threshold.

Laplacian energy of partial complement of a graph

The energy of the graph had its genesis in 1978. It is the sum of absolute values of its eigenvalues. It originates from the π -electron energy in the Huckel molecular orbital model but has also gained purely mathematical interest. Suppose μ1,μ2,…,μn is the Laplacian eigenvalues of G. The Laplacian energy of G has recently been defined as LE(G)=∑i=1nμi-2mn. In this paper, we define Laplacian energy of partial complements of a graph. Laplacian energy and spectrum of partial complements of the few classes of graphs are established. Some bounds and properties of Laplacian energy are obtained.

One-loop partition function, gauge accessibility and spectra in AdS3 gravity

We continue the study of the one-loop partition function of AdS3 gravity with focus on the square-integrability condition on the fluctuating fields. In a previous work we found that the Brown-Henneaux boundary conditions follow directly from the L2 condition. Here we rederive the partition function as a ratio of Laplacian determinants by performing a suitable decomposition of the metric fluctuations. We pay special attention to the asymptotics of the fields appearing in the partition function. We also show that in the usual computation using ghost fields for the de Donder gauge, such gauge condition is accessible precisely for square-integrable ghost fields. Finally, we compute the spectrum of the relevant Laplacians in thermal AdS3, in particular noticing that there are no isolated eigenvalues, only essential spectrum. This last result supports the analytic continuation approach of David, Gaberdiel and Gopakumar. The purely essential spectra found are consistent with the independent results of Lee and Delay of the essential spectrum of the TT rank-2 tensor Lichnerowickz Laplacian on asymptotically hyperbolic spaces.

On Laplacian spectrum of unitary Cayley graphs

Abstract Let R be a commutative ring with unity 1 ≠ 0 and let R× be the set of all unit elements of R. The unitary Cayley graph of R, denoted by GR = Cay(R, R×), is a simple graph whose vertex set is R and there is an edge between two distinct vertices x and y of R if and only if x − y ∈ R×. In this paper, we determine the Laplacian and signless Laplacian eigenvalues for the unitary Cayley graph of a commutative ring. Also, we compute the Laplacian and signless Laplacian energy of the graph GR and its line graph.

The spectrum and metric dimension of Indu–Bala product of graphs

Given a connected graph [Formula: see text], the distance Laplacian matrix [Formula: see text] is defined as [Formula: see text], and the distance signless Laplacian matrix [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the transmission matrix of [Formula: see text] and [Formula: see text] is the distance matrix of [Formula: see text]. The Indu–Bala product of two graphs [Formula: see text] and [Formula: see text], denoted by [Formula: see text], was introduced in (G. Indulal and R. Balakrishnan, Distance spectrum of Indu–Bala product of graphs, AKCE Int. J. Graph Comb. 13(3) (2016) 230–234). In this paper, we first obtain the distance Laplacian spectrum of [Formula: see text] in terms of Laplacian spectra of [Formula: see text] and [Formula: see text]. We then obtain the distance signless Laplacian spectrum of [Formula: see text] in terms of signless Laplacian spectra of [Formula: see text] and [Formula: see text]. We construct pair of graphs which are distance Laplacian co-spectral as well as pair of graphs which are distance signless Laplacian co-spectral. We further find the metric dimension of [Formula: see text] in terms of metric dimensions of [Formula: see text] and [Formula: see text]. Finally, we provide a problem for future research.

On a new functional for extremal metrics of the conformal Laplacian in high dimensions

In this paper, we introduce a new functional for the conformal spectrum of the conformal Laplacian on a closed manifold [Formula: see text] of dimension at least 3. For this new functional we provide a Korevaar type result. The main body of the paper deals with the case of the sphere but a section is devoted to more general closed manifolds. The functional introduced here has been investigated in the recent work [S. Perez-Ayala, Extremal eigenvalues of the conformal Laplacian under Sire–Xu normalization, preprint (2019); arXiv:2011.06018].

On the Laplacians for Strong Product Graphs Based on Polyacene Graphs

Let Hn be the molecular graph of a linear polyacene chain. In this paper, we introduce a kind of linear polyacene cylinder graph which is the strong product of two automorphism polyacene graphs Hn and Next, we obtain that the Laplacian spectrum of consists of the eigenvalues of symmetric matrices LA and LS of order according to the decomposition theorem of Laplacian characteristic polynomial. Furthermore, by applying the relationship between the roots and coefficients of the characteristic polynomials of the above two matrices, we study the Kirchhoff index and the number of spanning trees, and derive their explicit expressions in terms of n for the graph

On the Multiplicative Degree-Kirchhoff Indices and the Number of Spanning Trees of Linear Phenylene Chains

Let denote a molecular graph of linear phenylene, containing n hexagons and squares. In this paper, using the decomposition theorem of the normalized Laplacian characteristic polynomial, we obtain that the normalized Laplacian spectrum of consisting of the eigenvalues of two symmetric tridiagonal matrices of order 4n is determined. By applying the relationship between the roots and coefficients of the characteristic polynomial of the above two matrices, an explicit closed-form formula of the multiplicative degree-Kirchhoff index (resp. the number of spanning trees) of is derived.

Universal adjacency spectrum of zero divisor graph on the ring and its complement

For a commutative ring R with unity, the zero divisor graph is an undirected graph with all non-zero zero divisors of R as vertices and two distinct vertices u and v are adjacent if and only if uv = 0. For a simple graph G with the adjacency matrix A and degree diagonal matrix D, the universal adjacency matrix is where I is identity matrix and J is all-ones matrix. For a graph H on k vertices and a family of vertex disjoint regular graphs we determine eigenpairs of the universal adjacency matrix of H-join of in terms of eigenpairs of the adjacency matrix of Hi , and a symmetric matrix of order k. For a non-prime integer n > 3, we obtain eigenpairs of and As an application, we also discuss the adjacency, Seidel, Laplacian and signless Laplacian spectra of both and Lastly, we determine the characteristic polynomial of for prime p and integer m > 1 (except for with ).