Edge Neighborhoods in Line Graphs

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. Objects which possess certain properties of symmetry have been widely studied in mathematics, partly because of esthetic reasons. Their characterizations often demonstrate the strength of mathematical theories. The current paper deals with one kind of them, e-locally homogeneous graphs, characterizes those of them which are line graphs, and reveals their connections to other intensively studied areas of combinatorics. The terminology is based on [2]. Our graphs are finite, undirected, without loops and multiple edges. If v is a node, then N (v) means the set of nodes adjacent to v. By the neighborhood N (v) of a node v we mean the subgraph induced by N (v) and by N k (v) we mean the set of nodes with the distance k to v. A graph is said to be a locally-H graph, or a locally homogeneous graph, if for all its nodes v, N (v) is isomorphic to a given graph H.