Line-graph inertia imbalance

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.

Background

The authors establish a weaker bound n+(L(G))≤min{3n-(L(G)), n-(L(G))(n-(L(G))+1)/2} and prove the conjecture for trees and dense line graphs. Computational tests found no counterexamples for numerous graphs, but the conjecture remains open when the original graph has order n and size m with n≤m≤2n−2.

References

Computational investigation suggests that the upper bound for $n+$ of line graphs can be further improved. \begin{conjecture}\label{conj:line_graph} For any connected graph G , with line graph L(G) , n+(L(G)) \leq n-(L(G)) + 1. Equivalently, $s(L(G))\le 1$. \end{conjecture}

— New conjectures on the inertia of graphs  (2508.01163 - Akbari et al., 2 Aug 2025) in Conjecture 5.1, Section 5.1