Due to the limitation of energy resources, energy efficiency is a key issue in wireless sensor networks (WSNs). Clustering is proved to be an important way to realize hierarchical topology control, which can improve the scalability and prolong the lifetime of wireless sensor networks. In this paper, an energy-driven unequal clustering protocol (EDUC) for heterogeneous wireless sensor networks is proposed. EDUC includes an unequal clustering algorithm and an energy-driven adaptive cluster head rotation method. The unequal size of clusters can balance the energy consumption among clusters, and the energy-driven cluster head rotation method can achieve the balance of energy consumption among nodes within a cluster, which reduces the waste of energy. Simulation experiments show that EDUC balances the energy consumption well among the cluster heads and prolongs the network lifetime.
Let σk（G） denote the minimum degree sum of k independent vertices in G and α（G） denote the number of the vertices of a maximum independent set of G. In this paper we prove that if G is a 4-connected graph of order n and σ5（G） 〉 n ＋ 3σ（G） ＋ 11, then G is Hamiltonian.
