Let G = （V（G）, E（G）） be a simple connected graph of order n. For any vertices u, v, w ￠V（G） with uv ∈ E（G） and uw ￠ E（G）, an edge-rotating of G means rotating the edge uv （around u） to the non-edge position uw. In this work, we consider how the least eigenvalue of a graph perturbs when the graph is performed by rotating an edge from the shorter hanging path to the longer one.
The classical hypercube structure is a popular topological architecture in parallel computing environments and a large number of variations based on the hypercube were posed in the past three decades. Reliability evaluation of systems is important to the design and maintenance of multiprocessor systems. The h-extra edge-connectivity of graph G（V, E） is a kind of measure for the reliability of interconnection systems, which is defined as the minimum cardinality of a subset of edge set, if any, whose deletion disconnects G and such that every re- maining component has at least h vertices. This paper shows that the h-extra edge-connectivity 2n-1 2n-1 of the hypercube Qn is a constant 2n-1 for 2n-1/3≤ h2n-1, and n ≥ 4, which extends the result of [Bounding the size of the subgraph induced by m vertices and extra edge-connectivity of hypercubes, Discrete Applied Mathematics, 2013, 161（16）： 2753-2757].
