ABSTRACT
Many kinds of graph polynomials have been introduced and extensively studied, such as characteristic polynomial, chromatic polynomial, Tutte polynomial, matching polynomial, independence polynomial, and clique polynomial. Many properties of graph polynomials have been widely studied. This chapter surveys some results on the derivative and real roots of graph polynomials, which have applications in chemistry, control theory, and computer science. Related to the derivatives of graph polynomials, polynomial reconstruction of the matching polynomial is also introduced. Real roots of other graph polynomials have also been studied, such as edge-cover polynomial, the expected independence polynomial, domination polynomial, sigma polynomial, chromatic polynomial, Wiener polynomial, flow polynomial, and Tutte polynomial. Graph polynomials are polynomials assigned to graphs. Interestingly, they also arise in many areas outside graph theory as well. Many properties of graph polynomials have been widely studied.
