Jun Ge

Bio: Jun Ge is an academic researcher from Sichuan Normal University. The author has contributed to research in topics: Spanning tree & Bipartite graph. The author has an hindex of 5, co-authored 26 publications receiving 100 citations. Previous affiliations of Jun Ge include Soochow University (Suzhou) & Xiamen University.

Spectral radius and Hamiltonian properties of graphs

Abstract: Let be a graph with minimum degree . The spectral radius of , denoted by , is the largest eigenvalue of the adjacency matrix of . In this note, we mainly prove the following two results.(1) Let be a graph on vertices with . If , then contains a Hamilton path unless .(2) Let be a graph on vertices with . If , then contains a Hamilton cycle unless . As corollaries of our first result, two previous theorems due to Fiedler and Nikiforov and Lu et al. are obtained, respectively. Our second result refines another previous theorem of Fiedler and Nikiforov.

On Deploying Wireless Sensors to Achieve Both Coverage and Connectivity

Abstract: In this paper, we discuss some possible deployment patterns of wireless sensors to achieve both coverage and connectivity which were ignored in the discussion of "Deploying Wireless Sensors To Achieve Both Coverage And Connectivity, ACM, MobiHoc, 2006", and we prove that the strip-based patterns are absolutely optimal among all the deployment patterns of wireless sensors to achieve both coverage and 1-or 2-connectivity.

Spectral radius and Hamiltonian properties of graphs, II

Abstract: In this paper, we first present spectral conditions for the existence of $C_{n-1}$ in graphs (2-connected graphs) of order $n$, which are motivated by a conjecture of Erdős. Then we prove spectral conditions for the existence of Hamilton cycles in balanced bipartite graphs. This result presents a spectral analog of Moon-Moser's theorem on Hamilton cycles in balanced bipartite graphs, and extends a previous theorem due to Li and the second author for $n$ sufficiently large. We conclude this paper with two problems on tight spectral conditions for the existence of long cycles of given lengths.

Minimal sufficient sets of colors and minimum number of colors

Abstract: In this paper we first investigate minimal sufficient sets of colors for p=11 and 13. For odd prime p and any p-colorable link L with non-zero determinant, we give alternative proofs of mincol_p L \geq 5 for p \geq 11 and mincol_p L \geq 6 for p \geq 17. We elaborate on equivalence classes of sets of distinct colors (on a given modulus) and prove that there are two such classes of five colors modulo 11, and only one such class of five colors modulo 13. Finally, we give a positive answer to a question raised by Nakamura, Nakanishi, and Satoh concerning an inequality involving crossing numbers. We show it is an equality only for the trefoil and for the figure-eight knots.

Deploying wireless sensors to achieve both coverage and connectivity

Abstract: It is well-known that placing disks in the triangular lattice pattern is optimal for achieving full coverage on a plane. With the emergence of wireless sensor networks, however, it is now no longer enough to consider coverage alone when deploying a wireless sensor network; connectivity must also be con-sidered. While moderate loss in coverage can be tolerated by applications of wireless sensor networks, loss in connectivity can be fatal. Moreover, since sensors are subject to unanticipated failures after deployment, it is not enough to have a wireless sensor network just connected, it should be k-connected (for k > 1 ). In this paper, we propose an optimal deployment pattern to achieve both full coverage and 2-connectivity, and prove its optimality for all values of rc/rs, where rc is the communication radius, and rs is the sensing radius. We also prove the optimality of a previously proposed deployment pattern for achieving both full coverage and 1-connectivity, when rc/rs

Classification of Wireless Sensor Networks Deployment Techniques

Abstract: One of the main design aspects of Wireless Sensor Networks (WSNs) is the deployment strategy of the sensors. In general, WSN deployment methods fall under two categories: planned deployment and random deployment. In this paper, we focus on planned deployment which is defined as selectively deciding the locations of the sensors to optimize one or more design objectives of the WSN under some given constraints. There have been a large number of studies which proposed algorithms for solving the planned deployment problem. In this paper, we present a novel classification of the algorithms proposed in the literature for planned deployment of WSNs, based on the mathematical approach used for modeling and solving the deployment problem. Four distinct mathematical approaches are presented: Genetic Algorithms, Computational Geometry, Artificial Potential Fields and Particle Swarm Optimization. For each approach, we provide a discussion of its background and basic mathematical foundation. We then review the algorithms which belong to each approach and provide a comparison between them in terms of their objectives, assumptions and performance. Based on our extensive survey, we discuss the strengths and limitations of the four approaches and compare them in terms of the different WSN design factors.

Spectral analogues of Erdős’ and Moon–Moser’s theorems on Hamilton cycles

Abstract: In 1962, Erdős gave a sufficient condition for Hamilton cycles in terms of the vertex number, edge number and minimum degree of graphs which generalized Ore’s theorem. One year later, Moon and Moser gave an analogous result for Hamilton cycles in balanced bipartite graphs. In this paper, we present the spectral analogues of Erdős’ theorem and Moon–Moser’s theorem, respectively. Let be the class of non-Hamiltonian graphs of order n and minimum degree at least k. We determine the maximum (signless Laplacian) spectral radius of graphs in (for large enough n), and the minimum (signless Laplacian) spectral radius of the complements of graphs in . All extremal graphs with the maximum (signless Laplacian) spectral radius and with the minimum (signless Laplacian) spectral radius of the complements are determined, respectively. We also solve similar problems for balanced bipartite graphs and the quasi-complements.

Spectral radius and Hamiltonicity of graphs with large minimum degree

Abstract: Let G be a graph of order n and λ(G) the spectral radius of its adjacency matrix. We extend some recent results on sufficient conditions for Hamiltonian paths and cycles in G. One of the main results of the paper is the following theorem