Derive explicit penultimate-root bounds for vertex multiplications of T

Derive explicit lower bounds on the penultimate eigenvalue of the characteristic polynomial of every vertex multiplication of the 11-vertex graph T, potentially using further structural simplifications or the automorphism group of T.

Background

The paper reduces the simple-minimizer case to vertex multiplications of an explicit 11-vertex graph T and records the characteristic polynomial in the appendix. However, the resulting polynomial is sufficiently complicated that the authors do not obtain explicit bounds on its penultimate root.

The authors note two possible avenues for resolving this issue: eliminating the irrelevant eleventh vertex weight, and exploiting the automorphism group of the underlying simple graph, which is isomorphic to V₄ ≅ C₂ × C₂. Such bounds would strengthen the reduction from general weighted minimizers to the finite template T.