Spectral graph theory

About: Spectral graph theory is a research topic. Over the lifetime, 1334 publications have been published within this topic receiving 77373 citations.

Spectral clustering of combinatorial fullerene isomers based on their facet graph structure

Abstract: After Curl, Kroto and Smalley were awarded 1996 the Nobel Prize in chemistry, fullerenes have been subject of much research. One part of that research is the prediction of a fullerene's stability using topological descriptors. It was mainly done by considering the distribution of the twelve pentagonal facets on its surface, calculations mostly were performed on all isomers of $C_{40}, C_{60}$ and $C_{80}$. This paper suggests a novel method for the classification of combinatorial fullerene isomers using spectral graph theory. The classification presupposes an invariant scheme for the facets based on the Schlegel diagram. The main idea is to find clusters of isomers by analyzing their graph structure of hexagonal facets only. We also show that our classification scheme can serve as a formal stability criterion, which became evident from a comparison of our results with recent quantum chemical calculations. We apply our method to classify all isomers of $C_{60}$ and give an example of two different cospectral isomers of $C_{44}$. Calculations are done with MATLAB. The only input for our algorithm is the vector of positions of pentagons in the facet spiral. These vectors and Schlegel diagrams are generated with the software package Fullerene.

A new topological index for capacity allocation problem in survivable networks

Abstract: In this paper, we propose a new topological index, which is a numerical descriptor that characterizes survivable network topologies. A monotonically decreasing power law relationship can be found between this index and the total capacity allocation in the network. The new topological index is calculated based on the algebraic connectivity, which is adopted from spectral graph theory, more specifically it is based on the second-smallest eigenvalue of the Laplacian matrix of the network topology. Instead of the average nodal degree index that is usually used to characterize network connectivity in studies of the capacity allocation problem, our results suggest that this new topological index more accurately predicts the total capacity and is more informative. It can be used in studies on quantitative structure-performance relationships, in which the network performance or other properties of network are correlated with their topological structure. Initial studies of selected network topologies confirm that the connections between the total capacity and its network structure can be well described by this topological index in network survivability studies.

Spectrum of the Laplacian matrix of non-commuting graph of dihedral group D2n

Abstract: Let G be a graph with vertex set V = {v1,v2,..., vp}, A(G) is adjacency matrix of G and D(G) is diagonal matrix with entry dii = deg(vi), i = 1, 2, …, p. The Laplacian matrix of G is L(G) = D(G) – A(G). Spectrum of the Laplacian matrix is obtained by finding of eigenvalues of L(G) and their multiplicities. In this paper we study spectrum of the Laplacian matrix of non-commuting graph of dihedral group , and give results about characteristics polyniomial of L(T(D2n)) and its spectrum of the Laplacian matrix. We obtained spectrum of the Laplacian matrix of is SpecL(T(D2n))= [2n-1,n,0;n,n-2,1]

Sparse eigenvectors of graphs

Abstract: In order to analyze signals defined over graphs, many concepts from the classical signal processing theory have been extended to the graph case. One of these concepts is the uncertainty principle, which studies the concentration of a signal on a graph and its graph Fourier basis (GFB). An eigenvector of a graph is the most localized signal in the GFB by definition, whereas it may not be localized in the vertex domain. However, if the eigenvector itself is sparse, then it is concentrated in both domains simultaneously. In this regard, this paper studies the necessary and sufficient conditions for the existence of 1, 2, and 3-sparse eigenvectors of the graph Laplacian. The provided conditions are purely algebraic and only use the adjacency information of the graph. Examples of both classical and real-world graphs with sparse eigenvectors are also presented.

On the spectral invariants of symmetric matrices with applications in the spectral graph theory

Abstract: We first prove a formula which relates the characteristic polynomial of a matrix (or of a weighted graph), and some invariants obtained from its principal submatrices (resp. vertex deleted subgraphs). Consequently, we express the spectral radius of the observed objects in the form of power series. In particular, as is relevant for the spectral graph theory, we reveal the relationship between spectral radius of a simple graph and its combinatorial structure by counting certain walks in any of its vertex deleted subgraphs. Some computational results are also included in the paper.