A new family of bridged and twisted hypercubes

We show that by adding eight extra edges, referred to as bridges, to an n-cube (n⩾4) its diameter can be reduced by 2, and by adding sixteen bridges to an n-cube (n⩾6) its diameter can be reduced by 3. We also show that by adding (_m+1^4m)+1(m⩾2) bridges to an n-cube (n⩾4m and n⩾8) its diameter can be reduced by 2m and by adding 2(_m^4m-3)+1, (m>2) to an n-cube (n⩾4m-2 and n⩾10) its diameter can be reduced by 2m-1. We also consider the reduction of diameter of an n-cube by exchanging some independent edges (twisting), where two edges are called independent if they are not incident on a common node. We have shown that by exchanging four pairs of independent edges in a d-cube (d⩾5), we can reduce its diameter by 2. By exchanging sixteen pairs of independent edges, the diameter of a d-cube (d⩾7) can be reduced by 3. By exchanging 57 pairs of independent edges, the diameter can be reduced by 4 for d⩾9. To reduce the diameter by lower bound [d/2], (d⩾10) we need to exchange (_r+1^d-1) pairs of independent edges, where r=lower bound [d/4]+1