Graphs Whose Certain Polynomials Have Few Distinct Roots

Abstract

Let= ( , ) be a simple graph. Graph polynomials are a well-developed area useful for analyzing properties of graphs. We consider domination polynomial, matching polynomial, and edge cover polynomial of G. Graphs which their polynomials have few roots can sometimes give surprising information about the structure of the graph. This paper is primarily a survey of graphs whose domination polynomial, matching polynomial, and edge cover polynomial have few distinct roots. In addition, some new unpublished results and questions are concluded.

Some families of graphs with no nonzero real domination roots

“…As examples, the complete graph K n for odd n and the complete bipartite graph K n,n for even n, are in CG. With these motivation, in [1,5] the authors asked the question: "Which graphs have no nonzero real domination roots? "…”

“…The algebraic encoding of salient counting sequences allows one to not only develop formulas more easily, but also, often, to prove unimodality results via the nature of the the roots of the associated polynomials (a well known result of Newton states that if a real polynomial with positive coefficients has all real roots, then the coefficients form a unimodal sequence (see, for example, [16]). A root of D(G, x) is called a domination root of G (see [5,14]). The set of distinct non-zero roots of D(G, x) is denoted by Z * (D(G, x)).…”

On the domination polynomials of friendship graphs

“…The algebraic encoding of salient counting sequences allows one to not only develop formulas more easily, but also to prove often unimodality results via the nature of the the roots of the associated polynomials (a well known result of Newton states that if a real polynomial with positive coefficients has all real roots, then the coefficients form a unimodal sequence (see, for example, [19]). A root of D(G, x) is called a domination root of G. The set of distinct roots of D(G, x) is denoted by Z(D(G, x)) (see [4,6,17]).…”

Recurrence Relations and Splitting Formulas for the Domination Polynomial

“…We show in Theorem 2.4 that D(G, x) does not satisfy any linear recurrence relation which applies only the commonly used vertex operations of deletion, extraction, contraction and neighborhood-contraction. Nor does D(G, x) satisfy any linear recurrence relation using only edge deletion, contraction and extraction.In spite of this non-existence result, we give in this paper an abundance of recurrence relations and splitting formulas for the domination polynomial.The domination polynomial was studied recently by several authors, see [1,2,3,5,6,7,8,9,10,12]. The previous research focused mainly on the roots of domination polynomials and on the domination polynomials of various classes of special graphs.…”

“…The domination polynomial was studied recently by several authors, see [1,2,3,5,6,7,8,9,10,12]. The previous research focused mainly on the roots of domination polynomials and on the domination polynomials of various classes of special graphs.…”