A k-injective-edge coloring of a graph G is an edge coloring c: \(E(G)\rightarrow \{1,2,\cdots ,k\}\) , such that if \(e_{1}\) , \(e_{2}\) and \(e_{3}\) are three consecutive edges in G, then \(c(e_{1})\ne c(e_{3})\) , where \(e_{1}\) , \(e_{2}\) and \(e_{3}\) are consecutive if they form a path or a cycle of length 3. The minimum integer k such that G has a k-injective-edge coloring is called the injective chromatic index of G, denoted by \(\chi '_{i}(G)\) . The maximum average degree of G, denoted by mad(G), is defined to be the maximum average over all subgraphs H of G. In this paper, I will consider the injective edge coloring of sparse graph G with maximum degree 5. I get that (1) if mad \((G)<\frac{31}{8}\) , then \(\chi '_{i}(G)\le 20\) ; (2) if mad \((G)<4\) , then \(\chi '_{i}(G)\le 21\) .
