Study On Roots Of Two Graph Polynomials

The graph polynomials mostly get inspiration from the chromatic polynomial.In 1912,Birkhoff introduced the chromatic polynomial in order to prove the four-color conjecture.Then,many scholars are interested in the graph polynomials.The roots of graph polynomials is one of the important research areas,including the real roots,density and so on.For example,Sokal showed that the chromatic roots are dense in the entire complex plane,and then Brown et al.obtained that the independence roots and the domination roots are dense in the complex plane.In this paper,we mainly study the roots of total domination polynomials and adjoint polynomials.The details are as follows:In the first part,we mainly investigate the distribution of roots of total domination polynomials.Firstly,by using the Beraha-Kahane-Weiss theorem,we show the limiting curves of the total domination roots on friendship graph.Next,we obtain that the total domination roots are dense in the entire complex plane.Then,we present the recurrence relations of total domination polynomials of the corona product and get the expressions of total domination polynomials of some graphs by definition.In the second part,we consider the distribution of roots of adjoint polynomials.After discussing the value of adjoint polynomials at-1 by using the method of Bencs,we obtain that the roots of adjoint polynomials on the claw-free graph are real.Finally,we show that the roots of adjoint polynomials are dense in the entire complex plane.