Characterizations of line graphs in signed and gain graphs
arXiv:2101.09677 · doi:10.1016/j.ejc.2021.103479
Abstract
We generalize three classical characterizations of line graphs to line graphs of signed and gain graphs: the Krausz's characterization, the van Rooij and Wilf's characterization and the Beineke's characterization. In particular, we present a list of forbidden gain subgraphs characterizing the class of gain-line graphs. In the case of a signed graph whose underlying graph is a line graph, this list consists of exactly four signed graphs. Under the same hypothesis, we prove that a signed graph is the line graph of a signed graph if and only if its eigenvalues are either greater than , or less than , depending on which particular definition of line graph is adopted.
25 pages, 10 figures. Accepted for publication in European Journal of Combinatorics