On the Roots of Connected Domination Polynomials

Abstract

We determine the closure of the connected domination roots. The main tool is a substitution formula for the lexicographic product with a complete graph, $D_c(G[K_n],x)=D_c(G,(x+1{)}^n-1)$, proved in Theorem~\ref{thm:cd}. Combined with two explicit families of seed roots---the real roots of the cycles $C_n$ and the real roots of the joins $C_m\vee C_n$ lying in $(-1,0)$---this formula gives the two main results: the closure of the real connected domination roots is $(-\infty ,0]$, and the closure of all connected domination roots is the whole complex plane. These are the connected domination analogues of the root-density theorems of Brown and Tufts and of Brown and Beaton for the ordinary domination polynomial.

arXiv comment17 pages, 1 figure