Delta Conjecture for Minimum Semidefinite Rank

Prove the Delta Conjecture asserting that for any graph G (in particular, for the simple graphs representing molecular structures in the proposed framework), the minimum semidefinite rank msr(G) satisfies msr(G) ≤ |G| − δ(G), where |G| denotes the number of vertices of G and δ(G) denotes the minimum degree of G.

Background

Within the proposed string-graph model of molecular geometry, each molecule is represented by a simple graph G(V, E). The authors reference the Delta Conjecture from graph theory, relating the minimum semidefinite rank msr(G) to the order |G| and minimum degree δ(G) of the graph.

They explicitly cite the conjectural inequality msr(G) ≤ |G| − δ(G) to connect structural graph parameters of molecular graphs to linear-algebraic quantities, referencing prior work by Díaz-Navarro for context.