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.