Characterizations of line graphs in signed and gain graphs

TL;DR

This paper extends classical characterizations of line graphs to signed and gain graphs, providing new forbidden subgraph lists and eigenvalue criteria for these generalized structures.

Contribution

It generalizes classical line graph characterizations to signed and gain graphs, including forbidden subgraph lists and eigenvalue conditions.

Findings

Four signed graphs characterize gain-line graphs.

Eigenvalues determine if a signed graph is a line graph.

Generalized characterizations unify classical and signed graph theory.

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 $-2$, or less than $2$, depending on which particular definition of line graph is adopted.