Inertia bound for line graphs

Prove that for every connected graph G, its line graph L(G) satisfies n^+(L(G)) ≤ n^-(L(G))+1, equivalently s(L(G)) ≤ 1, where s denotes the adjacency-spectrum signature.

Background

The paper establishes the weaker bound n^+(L(G)) ≤ min{3n^-(L(G)), n^-(L(G))(n^-(L(G))+1)/2} for line graphs. It also proves the desired inequality for line graphs of trees and for sufficiently dense line graphs.

Computational tests found no counterexample among numerous graphs, including all graphs with at most nine vertices and line graphs with at most 100 vertices. The conjecture is known to be tight for several examples, including odd cycles of length 1 modulo 4, but remains unresolved for the remaining sparse range n(G) ≤ m(G) ≤ 2n(G)−2.