In the terminology used by Beineke (1968, 1970), the derived graph of a graph is its line
graph,
whose vertices are the edges of , with two vertices adjacent when the corresponding edges of
have a common endpoint.
In voltage graph theory, the derived graph of a voltage graph is the graph cover obtained from the associated
voltage assignment, meaning the graph whose vertices and edges lie over the vertices
and edges of the base graph (Gross 1974, Gross and
Tucker 1987).
Beineke, L. W. "Derived Graphs and Digraphs." In Beiträge zur Graphentheorie (Ed. H. Sachs, H. Voss, and
H. Walther). Leipzig, Germany: Teubner, pp. 17-33, 1968.Beineke,
L. W. "Characterizations of Derived Graphs." J. Combin. Th.9,
129-135, 1970.Gross, J. L. "Voltage Graphs." Disc.
Math.9, 239-246, 1974.Gross, J. L. and Tucker, T. W.
Topological
Graph Theory. New York: Wiley, 1987.
is its line
graph 
,
whose vertices are the edges of 
, with two vertices adjacent when the corresponding edges of
have a common endpoint.
