On the Real-rootedness, invariance, and root bounds of τ-polynomials

This paper explores new theoretical perspectives on Formula: see text-polynomials in algebraic graph theory through three main contributions. First, we identify classes of Formula: see text-polynomials invariant under standard graph operations, suggesting structural preservation under vertex amalgamation and edge decomposition via recursive combinatorial approaches. We derive closed-form expressions for Formula: see text-polynomials of cyclically glued graphs Formula: see text and Formula: see text using Stirling number expansions as representative cases. Second, we propose a root bounding framework based on Pringsheim’s theorem. Under specific graph density constraints, our analysis indicates that the maximal real root Formula: see text satisfies Formula: see text, where Formula: see text denotes the maximum degree. Third, we address the real-rootedness conjecture for complete graphs by developing an inductive proof structure. Through multiplier sequence analysis, we present evidence that Formula: see text admits exclusively real roots.