SCHULTZ POLYNOMIALS OF COMPOSITE GRAPHS

For a connected graph $G$, the {sc Schultz} and modified {sc Schultz} polynomials, introduced by {sc I. Gutman:} {it Some relations between distance-based polynomials of trees.} Bulletin, Classe des Sciences Math'ematiques et Naturelles, Sciences math'{e}matiques, Vol. CXXXI, extbf{30} (2005) 1--7, are defined as$H_1(G,x)=fraccc{1}{2}sum { (delta_u+ delta_v) x^{d(u,v|G)}mid u,v in V(G), u eq v }$ and $H_2(G,x)=fraccc{1}{2}sum{ (delta_u delta_v) x^{d(u,v| G)}mid u,v in V(G), u eq v}$, respectively, where $delta_u$ is the degree of vertex $u$,$d(u,v| G)$ is the distance between $u$ and $v$ and $V(G)$ is thevertex set of $G$. In this paper we find identities for the{sc Schultz} and modified {sc Schultz} polynomials of the sum, join and composition of graphs. As an application of our results we findthe {sc Schultz} polynomial of $C_4$ nanotubes.