suppose $L(G)$ is line graph of $G$ and k-regular and is connected,prove that $G$ is regular graph or bipartite of two kind of degree?
I made some classification:
if $L(G)$ is k-regular and have $K+1$ vertices ,$G$ can be $K_{1,k+1}$(star)
if $L(G)$ is k-regular and $K=2q$(even) then always we can consider $K_{q+1,q+1}$ as $G$.
if $L(G)$ is k-regular and $K=2p+1$(odd) then we can consider $K_{p+1,p+2}$ as $G$.
in even case it is also regular and if it doesn't even so it is bipartite,it just remains to show that no other case happen!
can you help me to make it complete with this approach,
or if you think that there is better approach it will be great to make aware,thanks alot.
Hint. Choose two adjacent vertices $u,v$ of $G$. Suppose $u$ had degree $p$, and $v$ has degree $q$; then $p+q=k+2$. Consider two cases.
Case I. $p=q$
Use connectedness to show that $G$ is $p$-regular.
Case II. $p\ne q$
Use connectedness to show that every vertex of $G$ has degree $p$ or $q$. Since $L(G)$ is $k$-regular, each edge of $G$ joins a vertex of degree $p$ to a vertex of degree $q$. Since $p\ne q$, this means that $G$ is a bipartite graph.