Bounding the size of the subgraph induced by mm vertices and extra edge-connectivity of hypercubes

In this note, we show that a subgraph induced by mm (denote mm by ∑i=0s2ti, t0=[log2m]t0=[log2m] and ti=[log2(m−∑r=0i−12tr)] for i≥1i≥1) vertices of an nn-dimensional hypercube has at most ∑i=0sti2ti−1+∑i=0si⋅2ti edges. As an application, we determine the mm-extra edge-connectivity of hypercubes for m≤2[n2].