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The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is theleast integer $d$ such that $G$ has an vertex (edge) labeling with $d$ labelsthat is preserved only by the trivial automorphism. It is known that for everygraph $G$ we have $D'(G) \leq D(G) + 1$. The complete characterization offinite trees $T$ with $D'(T)=D(T)+ 1$ has been given recently.  In this note we show that if $G$ is a finite connected graph with at leastone cycle, then $D'(G)\leq D(G)$. Finally, we characterize all connected graphsfor which $D'(G) \leq D(G)$.

The distinguishing index of graphs with at least one cycle is not more than its distinguishing number