Near automorphisms of $G_{(n,m)}$

Let $G$ be a graph with vertex set $V(G)$, $f$ a permutation of $V(G)$. Define ${\delta }_f(G)=\mathrm{\mid }d(x,y)-d(f(x),f(y))\mathrm{\mid }$ and ${\delta }_f(G)=\mathrm{\Sigma }{\delta }_f(x,y)$, where the sum is taken over all unordered pair $x$, $y$ of distinct vertices of $G$. ${\delta }_f(x,U)=\mathrm{\Sigma }{\delta }_f(x,y)$, where $U\subseteq V(G)$ and $y\in U$. Let $\pi (G)$ denote the smallest positive value of ${\delta }_f(G)$ among all permutations of $V(G)$. A permutation $f$ with ${\delta }_f(G)=\pi (G)$ is called a near automorphisms of $G$\cite{HV}. In this paper, we define $G_{(n,m)}$ is a graph obtained from $K_n$ by add $t_i$ pendent vertices to $y_i$ which is a vertex of $K_n$, $i=1,\cdots ,m$, and we say $y_i$ is a c-pendent vertex of $G_{(}n,m)$. We determine $\pi (G_{(n,m)})$ and describe permutations $f$ of $G_{(n,m)}$ for which $\pi (G_{(n,m)})={\delta }_f(G_{(n,m)})$. Because $G_{(1,1)}$ is a star and it is easy, hence we let $n\ge 2$. Suppose $G_{(}n,m)$ has $m$ c-pendent vertices $\{y_1,\dots ,y_m\}$ and $y_i$ has $t_i$ pendent vertices($1\le t_1\le t_2\le \dots \le t_m$). For $m<n$ we have $$\pi (G_{(n,m)})=\{\begin{array}{lc}2n-4& n\le t_1+2,m=1\\ 2t_1& otherwise\end{array}$$ For $m=n$ we have $$\pi (G_{(n,n)})=\{\begin{array}{lc}4& t_1=1,t_2=2\\ 2t_1+2t_2& otherwise\end{array}$$