Exact Real-Root Counts and Asymptotics for Domination Polynomials of Book Graphs

For the book graph Bn obtained by identifying a common edge in n copies of C4, we determinethe complete real-zero multiset of its domination polynomial and the first-order asymptotic locationsof all nonzero real zeros. For even n ≥ 2, there is one simple zero below −2 and one in (− 12 , 0);for odd n ≥ 3, there are two simple zeros in (− 12 , 0). In both cases zero has multiplicity two. Theexceptional graph B1 = C4 has no nonzero real zero. The root count establishes Conjecture 5.4 ofRather with its small-index correction, while the asymptotics provide quantitative refinements