Abstract.
By an edge-neighborhood of an edge $f$ in a graph we mean the subgraph induced by nodes outside $f$ which are adjacent to some node on $f$. Connected graphs whose line graphs have the same edge-neighborhood of any edge are characterized. There are $P_4,$ stars, complete graphs and regular triangle-free graphs in which any two nodes with the distance two have the same number of common neighbors.
Compressed Postscript 
