Meyniel conjectured the cop number c(G) of any connected graph G on n vertices is at most C √ n for some constant C. In this paper, we prove Meyniel's conjecture in special cases that G has diameter at most 2 or G is a bipartite graph with diameter at most 3. For general connected graphs, we prove c(G) = O(√ log 2 n ), improving the best previously known upper-bound O( n ln n ) due to Chiniforooshan.

Let G be a random graph on the vertex set {1, 2, . . . , n} such that edges in G are determined by independent random indicator variables, while the probability pij for {i, j} being an edge in G is not assumed to be equal. Spectra of the adjacency matrix and the normalized Laplacian matrix of G are recently studied by Oliveira and Chung-Radcliffe. Let A be the adjacency matrix of G,Ā = E(A), and ∆ be the maximum expected degree of G. Oliveira first proved that almost surely A −Ā = O( √ ∆ ln n) provided ∆ ≥ C ln n for some constant C. Chung-Radcliffe improved the hidden constant in the error term using a new Chernoff-type inequality for random matrices. Here we prove that almost surely A −Ā ≤ (2 + o(1)) √ ∆ with a slightly stronger condition ∆ ≫ ln 4 n. For the Laplacian L of G, Oliveira and Chung-Radcliffe proved similar results L −L| = O( √ ln n/ √ δ) provided the minimum expected degree δ ≫ ln n; we also improve their results by removing the √ ln n multiplicative factor from the error term under some mild conditions. Our results naturally apply to the classic Erdős-Rényi random graphs, random graphs with given expected degree sequences, and bond percolation of general graphs. IntroductionGiven an n × n symmetric matrix M , let λ 1 (M ), λ 2 (M ), . . . , λ n (M ) be the list of eigenvalues of M in the non-decreasing order. What can we say about these eigenvalues if M is a matrix associated with a random graph G? Here M could be the adjacency matrix (denoted by A) or the normalized Laplacian matrix (denoted by L). Both spectra of A and L can be used to infer structures of G. For example, the spectrum of A is related to the chromatic number and the independence number. The spectrum of L is connected to the mixing-rate of random walks, the diameters, the neighborhood expansion, the Cheeger constant, the isoperimetric inequalities, the expander graphs, the quasi-random graphs. For more applications of spectra of the adjacency matrix and the Laplacian matrix, please refer to monographs [3,9].Spectra of adjacency matrices and normalized Laplacian matrices of random graphs were extensively investigated in the literature. For the Erdős-Rényi random graph model G(n, p), Füredi and Komlós [15] proved that almost surely λ n (A) = (1 + o(1))np and max i≤n−1 |λ 1 (A)| ≤ (2 + o(1)) np(1 − p) provided np(1 − p) ≫ log 6 n; similar results are proved for sparse random graphs [11,16] and general random matrices [10,15]. Alon, Krivelevich, and Vu [1] showed that with high probability the s-th largest eigenvalue of a random

Despite of the extreme success of the spectral graph theory, there are relatively few papers applying spectral analysis to hypergraphs. Chung first introduced Laplacians for regular hypergraphs and showed some useful applications. Other researchers treated hypergraphs as weighted graphs and then studied the Laplacians of the corresponding weighted graphs. In this paper, we aim to unify these very different versions of Laplacians for hypergraphs. We introduce a set of Laplacians for hypergraphs through studying high-ordered random walks on hypergraphs. We prove the eigenvalues of these Laplacians can effectively control the mixing rate of high-ordered random walks, the generalized distances/diameters, and the edge expansions.

Additive manufacturing (AM) technology has rapidly evolved with research advances related to AM processes, materials, and designs. The advantages of AM over conventional techniques include an augmented capability to produce parts with complex geometries, operational flexibility, and reduced production time. However, AM processes also face critical issues, such as poor surface quality and inadequate mechanical properties. Therefore, several post-processing technologies are applied to improve the surface quality of the additively manufactured parts. This work aims to document post-processing technologies and their applications concerning different AM processes. Various types of post-process treatments are reviewed and their integrations with AM process are discussed.

Due to the rapid development of precision manufacturing technology, much research has been conducted in the field of multisensor measurement and data fusion technology with a goal of enhancing monitoring capabilities in terms of measurement accuracy and information richness, thereby improving the efficiency and precision of manufacturing. In a multisensor system, each sensor independently measures certain parameters. Then, the system uses a relevant signal-processing algorithm to combine all of the independent measurements into a comprehensive set of measurement results. The purpose of this paper is to describe multisensor measurement and data fusion technology and its applications in precision monitoring systems. The architecture of multisensor measurement systems is reviewed, and some implementations in manufacturing systems are presented. In addition to the multisensor measurement system, related data fusion methods and algorithms are summarized. Further perspectives on multisensor monitoring and data fusion technology are included at the end of this paper.