An explicit formula for the harmonic indices and harmonic polynomials of carbon nanocones CNC k [n

Abstract

Let G be a simple molecular graph without directed and multiple edges and without loops. The vertex and edge-sets of G are denoted by V(G) and E(G), respectively. Suppose G is also a connected molecular graph and let u, v ŒV(G) be two vertices. The harmonic index H(G) of G is defined as the sum of the weights 2(du+dv )-1 of all edges in E(G), where dv is the degree of a vertex v in G which is defined as the number of vertices of G adjacent to v. The harmonic polynomial of G is defined as H(G, x) = ∑ e=uυϵE(G ) 2x (du+dv–1) and there is the following nice relation between these two notions . In this paper, we present an explicit formula for the harmonic indices and harmonic polynomials of carbon nanocones CNC k [n].