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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.
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TL;DR: In this paper, the log-concavity of a type of iteration sequence related to the $\alpha$-normal weighted incidence matrices was studied for computing the spectral radius of hypergraphs.
Abstract: Identifying graphs with extremal properties is an extensively studied topic in spectral graph theory. In this paper, we study the log-concavity of a type of iteration sequence related to the $\alpha$-normal weighted incidence matrices which is presented by Lu and Man for computing the spectral radius of hypergraphs. By using results obtained about the sequence and the method of some edge operations, we will characterize completely extremal k-graphs with the smallest spectral radius among bicyclic hypergraphs with given size.
16 May 2012
TL;DR: This paper calculates the energy of some grid based networks using the multiset of eigenvalues of a graph, which is the sum of the absolute values of its eigen values.
Abstract: Eigenvalues of a graph are the eigenvalues of its adjacency matrix. The multiset of eigenvalues is called the spectrum. There are many properties which can be explained using the spectrum like energy, connectedness, vertex connectivity, chromatic number, and perfect matching etc. So it is very useful to calculate the spectrum of any graph. The energy of a graph is the sum of the absolute values of its eigenvalues. In this paper we calculate the energy of some grid based networks.
Posted Content•
TL;DR: The associated transmission graph Laplacian is shown to have many of the former properties of the classical case, including an upper Cheeger type bound on the second eigenvalue minus the first of a geometric isoperimetric character.
Abstract: The standard notion of the Laplacian of a graph is generalized to the setting of a graph with the extra structure of a ``transmission`` system. A transmission system is a mathematical representation of a means of transmitting (multi-parameter) data along directed edges from vertex to vertex. The associated transmission graph Laplacian is shown to have many of the former properties of the classical case, including: an upper Cheeger type bound on the second eigenvalue minus the first of a geometric isoperimetric character, relations of this difference of eigenvalues to diameters for k-regular graphs, eigenvalues for Cayley graphs with transmission systems. An especially natural transmission system arises in the context of a graph endowed with an association. Other relations to transmission systems arising naturally in quantum mechanics, where the transmission matrices are scattering matrices, are made. As a natural merging of graph theory and matrix theory, there are numerous potential applications, for example to random graphs and random matrices.
TL;DR: It is pointed out that the bound is explicitly computable by using the valencies and the intersection matrix of a distance-regular graph by the discrete Green’s function, which is defined as the inverse of β-Laplacian for some positive real number β.
Abstract: We present an upper bound on the Cheeger constant of a distance-regular graph. Recently, the authors found an upper bound on the Cheeger constant of distance-regular graph under a certain restriction in their previous work. Our new bound in the current paper is much better than the previous bound, and it is a general bound with no restriction. We point out that our bound is explicitly computable by using the valencies and the intersection matrix of a distance-regular graph. As a major tool, we use the discrete Green’s function, which is defined as the inverse of β-Laplacian for some positive real number β. We present some examples of distance-regular graphs, where we compute our upper bound on their Cheeger constants.
Journal Article•
TL;DR: In this paper, the authors generalize the results of Wang, Li and Zhang to families of integral trees with diameters 4, 6 and 8, and show that these families can be constructed infinitely by solving some Diophantine equations.
Abstract: An integral graph is a graph of which all the eigenvalues of its adjacency matrix are integers. This paper investigates integral trees of diameter 4. Many new classes of such integral trees are constructed infinitely by solving some certain Diophantine equations. These results generalize some results of Wang, Li and Zhang (see Families of integral trees with diameters 4, 6 and 8, Discrete Applied Mathematics, 2004, 136: 349-362). Keywords Operations research, integral tree, characteristic polynomial, dio- phantine equation, graph spectrum Subject Classification (GB/T 13745-1992) 110.74