Geodesic
EDGE NEIGHBORHOODS IN LINE GRAPHS
Acta mathematica Universitatis Comenianae, Tome 62 (1993) no. 2 
L. Soltes. EDGE NEIGHBORHOODS IN LINE GRAPHS. Acta mathematica Universitatis Comenianae, Tome 62 (1993) no. 2. http://geodesic.mathdoc.fr/item/AMUC_1993_62_2_a2/
@article{AMUC_1993_62_2_a2,
     author = {L. Soltes},
     title = {EDGE {NEIGHBORHOODS} {IN} {LINE} {GRAPHS}},
     journal = {Acta mathematica Universitatis Comenianae},
     year = {1993},
     volume = {62},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/AMUC_1993_62_2_a2/}
}
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Voir la notice de l'article provenant de la source Comenius University

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.