on the strength and independence number of powers of paths and cycles

A numbering $f$ of a graph $g$ of order $n$ is a labeling that assigns distinct elements of the set $left{1,2, ldots, n right}$ to the vertices of $g$. the strength $mathrm{str}left(gright) $ of $g$ is defined by $mathrm{str}left( gright) =min left{ mathrm{str}_{f}left( gright)leftvert ftext{ is a numbering of }gright. right}$, where $mathrm{str}_{f}left( gright) =max left{ fleft( uright)+fleft( vright) leftvert uvin eleft( gright) right. right} $.using the concept of independence number of a graph, we determine formulas for the strength of powers of paths and cycles. to achieve the latter result, we establish a sharp upper bound for the strength of a graph in terms of its order and independence number and a formula for the independence number of powers of cycles.