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 is an element of V(G) be two vertices. The harmonic index H(G) of G is defined as the sum of the weights 2(d(u)+d(v))(-1) of all edges in E(G), where d(v) 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) = Sigma(e=uv is an element of E(G)) 2x((du+dv-1)) and there is the following nice relation between these two notions H(G) = integral(1)(0) H(G, x)dx. In this paper, we present an explicit formula for the harmonic indices and harmonic polynomials of carbon nanocones CNCk[n].

Eccentricity based topological indices have attracted large attraction in the field of chemical graph theory. Eccentricity based topological indices are prominent due to their wide range applications. With the help of eccentricity based topological indices we can predict excellent accuracy rate in certain biological activities of diverse nature as com-pared to other indices. Carbon nanotubes are the cylindrical molecules that consist of rolled up sheets of carbon atoms. In this paper we consider the chemical graph of NA(m)(n) nanotube and compute an important eccentricity based topological index called Modified eccentric connectivity index.

The resistance distance (Kirchhoff index and multiplicative Kirchhoff index) and distance-based (Wiener index and Gutman index) graph invariants of Γn = P2 ×Kn are considered. Firstly by using the decomposition theorem, we procure the Laplacian and Normalized Laplacian spectrum for graph Γn, respectively. Based on which, we can procured the formulae for the number of spanning trees and some resistance distance and distance-based graph invariants of graph Γn. Also, it is very interesting to see that when n tends to infinity, Kf (Γn) is a polynomial and W (Γn) is a quadratic polynomial.