Perfecting the Line Graph

TL;DR

This paper introduces a new graph construction called the doubled edge-stage lift, analyzing its spectral properties, perfectness, and structural features, with explicit examples like Paley lifts demonstrating its theoretical and practical significance.

Contribution

The paper presents a novel graph lift construction that preserves edge-space information, provides explicit spectral formulas, and yields families of perfect, claw-free, and box-perfect graphs.

Findings

The construction yields perfect, claw-free, and box-perfect graphs for all input graphs.

Explicit spectral formulas and bounds for the second eigenvalue are derived in the regular case.

Paley lifts produce explicit regular perfect graphs with controlled spectra.

Abstract

We study the doubled edge-stage lift \[ \HL'_2(G)=L(G\otimes K_2), \] the line graph of the canonical bipartite double cover of a graph \(G\). The natural involution \((u,v)\leftrightarrow(v,u)\) has quotient isomorphic to \(L(G)\), and induces a sector decomposition \[ \Spec(\HL'_2(G))=\Spec(L(G))\cup\Spec(\mathcal A(G)), \] where \(\mathcal A(G)\) is a canonical signed refinement of the line graph. Thus the construction retains substantial edge-space information through its quotient and antisymmetric sector. For every input graph, \(\HL'_2(G)\) is perfect, claw-free, and box-perfect. In the regular case we give an explicit spectral formula, together with quantitative control of the second eigenvalue and spectral gap for non-bipartite input. Explicit families, including the complete-graph lifts and the Paley lifts, illustrate the theory; in particular, the Paley lifts furnish an explicit family of regular perfect graphs with controlled adjacency spectrum and spectral gap. The construction may be viewed both intrinsically, via ordered-edge adjacency by one-coordinate agreement, and extrinsically, as the line graph of the canonical double cover. The first viewpoint emphasizes the edge-stage nature of the lift, while the second supplies the structural proofs used here.