International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 29, Pages 1509-1521
Old and new generalizations of line graphs
Department of Computer Science, Ball State University, Muncie 47306, IN, USA
Received 8 October 2003
Copyright © 2004 Jay Bagga. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Line graphs have been studied for over seventy years. In 1932, H.
Whitney showed that for connected graphs, edge-isomorphism
implies isomorphism except for and . The line
graph transformation is one of the most widely studied of all
graph transformations. In its long history, the concept has been
rediscovered several times, with different names such as derived
graph, interchange graph, and edge-to-vertex dual. Line graphs
can also be considered as intersection graphs. Several variations
and generalizations of line graphs have been proposed and
studied. These include the concepts of total graphs, path graphs,
and others. In this brief survey we describe these and some more
recent generalizations and extensions including super line graphs
and triangle graphs.