Schultz polynomials of composite graphs

Abstract

For a connected graph G, the Schultz and modified Schultz polynomials, introduced by I. Gutman: Some relations between distance-based polynomials of trees. Bulletin, Classe des Sciences Math́ematiques et Naturelles, Sciences math́ematiques, Vol. CXXXI, 30 (2005) 1-7, are defined as H1(G,x) = 1/2∑{(δu + δv)xd(u,v,|G) | u,v,∈ V(G), u ≠ v} and H2(G,x) = 1/2∑{(δuδv)xd(u,v,|G) | u,v,∈ V(G), u ≠ v}, respectively, where δu is the degree of vertex u, d(u, v|G) is the distance between u and v and V(G) is the vertex set of G. In this paper we find identities for the Schultz and modified Schultz polynomials of the sum, join and composition of graphs. As an application of our results we find the Schultz polynomial of C4 nanotubes.