Reduced-graph order bound by negative inertia

Prove that for every non-negative integer k, every reduced graph with exactly k negative adjacency eigenvalues has order at most 2^{k+1}−2.

Background

The paper presents Mohammadian’s conjecture as a sharp proposed bound for the order of reduced graphs with prescribed negative inertia. The bound is attained by a recursive construction originating with Kotlov and Lovász, beginning with K_2 and increasing both order and negative inertia through a specified graph operation.