Python Verification Script for: "On minimizing the signed Estrada index: A counterexample via book graphs"

This repository provides the official, standalone Python verification script (verify_table1.py) accompanying the mathematical research article: "On minimizing the signed Estrada index: A counterexample via book graphs". The article constructs a rigorous counterexample on the four-page book graph (B4) to disprove a recent open conjecture regarding the minimization of the signed Estrada index of connected graphs. This script provides an independent, exhaustive, and exact computational audit of all mathematical claims, characteristic polynomials, algebraic eigenvalues, and interval enclosures presented in Table 1 of the paper. Key Verification Features Exact Symbolic Algebra: Uses SymPy to verify the characteristic polynomial factorization for all admissible page-triangle sign sums S in {-4, -2, 0, 2, 4}. High-Precision Arithmetic: Uses mpmath (set to 80 decimal places of precision) to compute exact spectral radii (mu_1) and exact Estrada indices (EE), confirming that the minimum index occurs uniquely at S = -2, whereas the global minimum Estrada index occurs uniquely at S = -4. Matrix Exponential Cross-Check: Constructs the explicit 6x6 signed adjacency matrices for B4 and independently verifies the Estrada index against numerical matrix exponentials via scipy.linalg.expm and numpy.linalg.eigvalsh. Certified Rational Enclosures: Implements an exact rational Taylor-remainder certificate (with Maclaurin degree N = 24) over the ordered field of rational numbers (Q), certifying that the true mathematical values are strictly enclosed within the interval bounds reported in Table 1 without floating-point rounding errors. Exhaustive Space Enumeration: Conducts a brute-force check across all 2^9 = 512 raw edge-sign configurations of B4, verifying that every raw signature shares the exact spectrum of its corresponding normal-form representative. System Requirements and Dependencies To run this verification script locally, ensure you have Python 3.8+ installed along with the following standard scientific libraries: pip install sympy numpy scipy mpmath Usage Execute the script directly from the terminal or command prompt: python verify_table1.py The script will run all symbolic and numeric checks seamlessly and print a detailed verification report to the standard output, concluding with: All checks passed.