The edit distance function of a hereditary property $mathscr{H}$ is the asymptotically largest edit distance between a graph of density $pin[0,1]$ and $mathscr{H}$. Denote by $P_n$ and $C_n$ the path graph of order $n$ and the cycle graph of order $n$, respectively. Let $C_{2n}^*$ be the cycle graph $C_{2n}$ with a diagonal, and $widetilde{C_n}$ be the graph with vertex set ${v_0, v_1, ldots, v_{n-1}}$ and $E(widetilde{C_n})=E(C_n)cup {v_0v_2}$. Marchant and Thomason determined the edit distance function of $C_6^{*}$. Peck studied the edit distance function of $C_n$, while Berikkyzy et al. studied the edit distance of powers of cycles. In this paper, by using the methods of Peck and Martin, we determine the edit distance function of $C_8^{*}$, $widetilde{C_n}$ and $P_n$, respectively.
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