Pandey’s real-rootedness parity conjecture for generalized Petersen graphs

For a generalized Petersen graph GP(n,k), n≥2k+1, the independence polynomial counts independent vertex sets by size. Pandey conjectured that all its roots are real exactly when k is even.

Why it matters and what counts as progress

Why it matters: This tests whether a simple parity condition determines real-rootedness in an entire graph family.

What counts as progress: A proof of either implication, or exact polynomial counterexamples with valid n,k and rigorously determined root behavior.

How the problem developed

5 Jan 2026 · Conjecture [1]
Rohan Pandey Used exact transfer-matrix computations for k≤4 and n≤30 to conjecture that the independence polynomial of GP(n,k) is real-rooted exactly when k is even.
8 Jul 2026 · Counterexample [2][3]
Erlbacher and Demonstrandum Reported exact examples refuting both directions: GP(9,2) is not real-rooted despite even k, while GP(7,3) and GP(3,1) are real-rooted despite odd k.

AI contribution

Exact graph checks refute both directions of Pandey’s parity conjecture

GP(9,2) has a non-real-rooted independence polynomial although k is even. GP(7,3) and GP(3,1) are real-rooted although k is odd. These examples refute both directions of Pandey’s claim for generalized Petersen graphs.

AI involvementAI-centered

ImportanceModest

Technical verificationFormal (exact computation)

Human understandingFew