Lower Bounds on Expansions of Graph Powers

Given a lazy regular graph G, we prove that the expansion of G^t is at least sqrt(t) times the expansion of G. This bound is tight and can be generalized to small set expansion. We show some applications of this result

Co-designed 3+3 port dual-band broadside tri-modal patch antenna

A co-designed (3+3)-port antenna for dual-band operation that requires no feeding or decoupling network is proposed and verified. The antenna consists of six ports which are divided into two groups that resonate in two differentfrequency ranges. The basic radiating element is a tri-modal patch in a folded snowflake-shaped structure with the largest antenna dimension of 0.48λ1 or 0.68λ2, where λ1 and λ2 are the wavelengths in air at the center frequencies of the lower and upper operating bands, respectively. Measurement results showthat the 10 dB impedance bandwidths of the two respective bands are 19.7% and 14.4%. The proposed antenna exhibits compact, multiport, multiband and broadside radiation characteristics which are not only suitable for dual-band MIMO applications, but also for energy harvesting systems with spatial and frequency diversities, or dual-function wireless systems with simultaneousinformation and power transfer

Bandwidth enhancement technique for broadside tri-modal patch antenna

A technique for enhancing the bandwidth of a broadside tri-modal patch antenna is described. The key idea of the technique is to incorporate a dual-resonance structure into the broadside tri-modal patch geometry. By increasing one edge of the tri-modal patch while decreasing its size at the opposite edge, the resulting structure can be viewed as two super-imposed Y-shaped structures of different resonant frequencies. This intuition is confirmed using characteristic mode analysis (CMA). Furthermore, guided by CMA, further modificationsenable two sets of resonant modes to be tuned for increasing the bandwidth of the tri-modal patch antenna. Importantly, the proposed bandwidth enhancement technique does not affect the desired broadside radiation patterns significantly. Therefore, it can be utilized to modify the tri-modal patch antenna withoutdegrading its potential for massive MIMO array application. Measurement results show that the technique enhances the 10 dB impedance bandwidth from 4.3% to 19.7% with the largest antenna dimension of 0.48λc, where λc is the wavelength in air at the center frequency. The design example of the proposed technique is able to cover widely used 3 GHz bands in 5G communication systems and its potential usage in massive MIMO arrays is demonstrated

Two-port design of Y-shaped patch using characteristic mode analysis

The design of two antenna ports for a compact Y-shaped patch on top of a square ground plane is considered in this work. Specifically, characteristic mode analysis reveals candidate regions to locate two direct probe feeding ports to jointly excite two significant modes of the Y-shaped patch with low mutual coupling and correlation. The feed design procedure, based on both the amplitude and phase distributions of the characteristic electric near-fields, is validated in full-wave simulation with two example realizations of two-port designs

Improved Cheeger's Inequality: Analysis of Spectral Partitioning Algorithms through Higher Order Spectral Gap

Let \phi(G) be the minimum conductance of an undirected graph G, and let 0=\lambda_1 <= \lambda_2 <=... <= \lambda_n <= 2 be the eigenvalues of the normalized Laplacian matrix of G. We prove that for any graph G and any k >= 2, \phi(G) = O(k) \lambda_2 / \sqrt{\lambda_k}, and this performance guarantee is achieved by the spectral partitioning algorithm. This improves Cheeger's inequality, and the bound is optimal up to a constant factor for any k. Our result shows that the spectral partitioning algorithm is a constant factor approximation algorithm for finding a sparse cut if \lambda_k$ is a constant for some constant k. This provides some theoretical justification to its empirical performance in image segmentation and clustering problems. We extend the analysis to other graph partitioning problems, including multi-way partition, balanced separator, and maximum cut

Cheeger Inequalities for Vertex Expansion and Reweighted Eigenvalues

The classical Cheeger's inequality relates the edge conductance $\phi$ of a graph and the second smallest eigenvalue $\lambda_2$ of the Laplacian matrix. Recently, Olesker-Taylor and Zanetti discovered a Cheeger-type inequality $\psi^2 / \log |V| \lesssim \lambda_2^* \lesssim \psi$ connecting the vertex expansion $\psi$ of a graph $G=(V,E)$ and the maximum reweighted second smallest eigenvalue $\lambda_2^*$ of the Laplacian matrix. In this work, we first improve their result to $\psi^2 / \log d \lesssim \lambda_2^* \lesssim \psi$ where $d$ is the maximum degree in $G$, which is optimal assuming the small-set expansion conjecture. Also, the improved result holds for weighted vertex expansion, answering an open question by Olesker-Taylor and Zanetti. Building on this connection, we then develop a new spectral theory for vertex expansion. We discover that several interesting generalizations of Cheeger inequalities relating edge conductances and eigenvalues have a close analog in relating vertex expansions and reweighted eigenvalues. These include an analog of Trevisan's result on bipartiteness, an analog of higher order Cheeger's inequality, and an analog of improved Cheeger's inequality. Finally, inspired by this connection, we present negative evidence to the $0/1$-polytope edge expansion conjecture by Mihail and Vazirani. We construct $0/1$-polytopes whose graphs have very poor vertex expansion. This implies that the fastest mixing time to the uniform distribution on the vertices of these $0/1$-polytopes is almost linear in the graph size. This does not provide a counterexample to the conjecture, but this is in contrast with known positive results which proved poly-logarithmic mixing time to the uniform distribution on the vertices of subclasses of $0/1$-polytopes.Comment: 65 pages, 1 figure. Minor change