A characterization of total graphs
Mehdi Behzad · Proceedings of the American Mathematical Society · 1970
We consider “ordinary” graphs; that is, finite undirected graphs with no loops or multiple edges. The total graph T ( G ) T(G) of a graph G G is that graph whose vertex set is V ( G ) ∪ E ( G ) V(G) \cup E(G) and in which two vertices are adjacent if and only if they are adjacent or incident in G G . A characterization of regular total graphs as well as some other properties of total graphs have been considered before. In this article we consider nonregular graphs and yield a method which enables us actually to determine whether or not they are total.