Let n 1 be an integer. The hypercube Qn is the graph whose vertex set isf0;1gn, where two n-tuples are adjacent if they differ in precisely one coordinate. This graph has many applications in Computer sciences and other area of sciences. Inthe graph Qn, the layer Lk is the set of vertices with exactly k 1’s, namely, vertices ofweight k, 1 k n. The hyper-star graph B(n;k) is...

Spectrum is a valuable, scarce and finite natural resource that is needed for many different applications, so efficient use of the scarce radio spectrum is important for accommodating the rapid growth of wireless communications. Spectrum auctions are one of the best-known market-based solutions to improve the efficiency of spectrum use. However, Spectrum auctions are fundamentally differen...

In this class, we introduced the random walk on graphs. The last lecture shows Perron-Frobenius theory to the analysis of primary eigenvectors which is the stationary distribution. In this lecture we will study the second eigenvector. To analyze the properties of the graph, we construct two matrices: one is (unnormalized) graph Laplacian and the other is normalized graph Laplacian. In the first...

Independent component analysis (ICA) of a mixed signal into a linear combination of its independent components, is one of the main problems in statistics, with wide range of applications. The un-mixing is usually performed by finding a rotation that optimizes a functional closely related to the differential entropy. In this paper we solve the linear ICA problem by analyzing the spectrum and eig...

The energy of a graph G is equal to the sum of absolute values of the eigenvalues of the adjacency matrix of G, whereas the Laplacian energy of a graph G is equal to the sum of the absolute value of the difference between the eigenvalues of the Laplacian matrix of G and average degree of the vertices of G. Motivated by the work from Sharafdini et al. [R. Sharafdini, H. Panahbar, Vertex weighted...

In this work we consider the nonlinear graph p-Laplacian and the set of eigenvalues and associated eigenvectors of this operator defined by a variational principle. We prove a unifying nodal domain theorem for the graph p-Laplacian for any p ≥ 1. While for p > 1 the bounds on the number of weak and strong nodal domains are the same as for the linear graph Laplacian (p = 2), the behavior changes...

In this paper, we established a connection between the Laplacian eigenvalues of a signed graph and those of a mixed graph, gave a new upper bound for the largest Laplacian eigenvalue of a signed graph and characterized the extremal graph whose largest Laplacian eigenvalue achieved the upper bound. In addition, an example showed that the upper bound is the best in known upper bounds for some cases.

The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the Laplacian matrix of the graph. In this paper, we investigate Laplacian spread of graphs, and prove that there exist exactly five types of tricyclic graphs with maximum Laplacian spread among all tricyclic graphs of fixed order.

For a graph, the least signless Laplacian eigenvalue is the least eigenvalue of its signless Laplacian matrix. This paper investigates how the least signless Laplacian eigenvalue of a graph changes under some perturbations, and minimizes the least signless Laplacian eigenvalue among all the nonbipartite graphs with given matching number and edge cover number, respectively.

For a graph, the least signless Laplacian eigenvalue is the least eigenvalue of its signless Laplacian matrix. This paper investigates how the least signless Laplacian eigenvalue of a graph changes under some perturbations, and minimizes the least signless Laplacian eigenvalue among all the nonbipartite graphs with given matching number and edge cover number, respectively.