The wide-diameter of regular graphs

The diameter with width m of a graph G is defined as the minimum integer d for which between any two distinct vertices in G there exist at least m internally disjoint paths of length of at most d. It was shown that the tight upper bound on m-diameter of w-regular w-connected graph with order n is(n-2)(w-2)(w-m+1)(3m-w-4)+1 for any integer m with 2w+53≤m≤w. Some known results can be deduced or improved from the obtained result.