The differential of the line graph L(G)

Abstract

Let G = ( V , E ) be a simple graph with vertex set V and edge set E . Let D be a subset of V , and let B ( D ) be the set of neighbors of D in V ∖ D . The differential ∂ ( D ) of D is defined as | B ( D ) | − | D | . The maximum value of ∂ ( D ) taken over all subsets D ⊆ V is the differential of G , denoted by ∂ ( G ) . The line graph L ( G ) of G = ( V , E ) is the graph whose vertex set is E , and two vertices in L ( G ) are adjacent if and only if their corresponding edges in G have a common end vertex. In this work we prove that ∂ ( L ( G ) ) ≥ ∂ ( G ) − 1 for any graph G , and that ∂ ( L ( G ) ) ≥ ∂ ( G ) for any graph G different from a tree. Moreover, we give a characterization of all trees T such that ∂ ( L ( T ) ) = ∂ ( T ) − 1 .