Perfecting the Line Graph

We study the doubled edge-stage lift $${\text{HL}}_2^{\mathrm{\prime }}(G)=L(G\otimes K_2),$$ 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 $$\text{Spec}({\text{HL}}_2^{\mathrm{\prime }}(G))=\text{Spec}(L(G))\cup \text{Spec}(\mathcal{A}(G)),$$ 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, ${\text{HL}}_2^{\mathrm{\prime }}(G)$ 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.