Minimum distance spectral radius for the remaining edge-size range

Prove that, for every integer me3, with n the unique positive integer satisfying (n−1 choose 2) < m ≤ (n choose 2), s = m − (n−1 choose 2), and 1 ≤ s ≤ (n−6)/2, every connected graph G with m edges satisfies ρ(G) ≥ ρ(P_{n,s+1}), with equality if and only if G is isomorphic to P_{n,s+1}, where P_{n,s+1} is the disjoint union of s+1 paths whose orders differ by at most one.

Background

The paper determines the connected graphs with minimum distance spectral radius among graphs with a fixed number m of edges for the range max{(n−6)/2, 1} ≤ s ≤ n−1, where n is determined by (n−1 choose 2) < m ≤ (n choose 2) and s = m − (n−1 choose 2). The candidate extremal graph is P_{n,s+1}, the disjoint union of s+1 paths with nearly equal orders; its complement is connected and has m edges.

The remaining case is 1 ≤ s ≤ (n−6)/2. Structural results in Theorem 1.2 show that a minimizing graph has order n and impose strong degree constraints, while Theorem 3.2 settles the case in which the complement is a forest. The authors report computational evidence for the first uncovered parameter cases and state that, if the conjecture holds, it would suffice to prove that every minimizing complement is a forest.