The answer to this question could be trivial!
The line graph of simple $d$-regular graph is ($2d-2$)-regular, since each edge is connected to $d-1$ edges for each of its two vertices.
My question is the line graph of regular multigraph is also regular graph?
I think that only the loops could effect the regularity of the line graph, but I'm not sure.
Any idea will be useful!
The degree of a vertex in the line graph corresponds to the number of edges incident to the corresponding edge. So in a $d$-regular-degree multigraph, if you have loops, then the loops see $d-1$ edges while the non-loops are adjacent to the edges of two vertices. So if there are loops and non-loops the line graph isn't regular.
But even if you consider loopless multigraphs, consider an edge $e$ adjacent to vertices $v_1$ and $v_2$. It is adjacent to $|e(v_1)\setminus e| + |e(v_2)\setminus e| - |(e(v_1) \cap e(v_2))\setminus e|$ edges, which by regularity is $(d-1) + (d-1) - |(e(v_1) \cap e(v_2))\setminus e|$; now you see that if there are pairs of vertices with different number of edges between them, the line graph isn't regular.
So the line graph of a $d$-regular multigraph is regular if it is loopless and every pair of vertices is linked by either $0$ or $k$ (for some $k<d$) edges.