ABC index, ABC4 index, Randic connectivity index, Sum connectivity index, GA index, GA5 index, harmonic index, second zagreb index and AZI of Fluorographene are computed.

ABC index, ABC4 index, Randic connectivity index, Sum connectivity index, GA index, GA5 index, harmonic index, second zagreb index and AZI of Fluorographene are computed.

The concept of energy of graph is defined as the sum of the absolute values of the eigenvalues of a graph. Let λ1, λ2, . . . , λn be eigenvalues of graph G, then the energy of G is defined as E (G) =∑nn=1|λه|. The aim of this paper is to compute the eigenvalues of two fullerene graphs C60 and C80.

In this paper, computation of the Arithmetic-Geometric index (AG1 index), SK index, SK1 index and SK2 index of H-Naphtalenic nanotube and TUC4[m,n] nanotube. We also compute SK3 index, AG2 index for H-Naphtalenic nanotube and TUC4[m,n] nanotube.

In this paper, we study the energy levels of an electron moving in a thin film. This film is considered as a two-dimensional electron gas which is under the influence of a uniform external magnetic field B and a uniform external electric field E. Here, the magnetic field is perpendicular to the film. Also, in this paper, we have selected the Landau gauge, because this gauge is useful for working in rectangular geometries.

Recently, Shigehalli and Kanabur [20] have put forward for new degree based topological indices, namely Arithmetic-Geometric index (AG1 index), SK index, SK1 index and SK2 index of a molecular graph G. In this paper, we obtain the explicit formulae of these indices for Polyhex Nanotube without the aid of a computer.

Let $G$ be a graph with a vertex weight $omega$ and the vertices $v_1,ldots,v_n$. The Laplacian matrix of $G$ with respect to $omega$ is defined as $L_omega(G)=diag(omega(v_1),cdots,omega(v_n))-A(G)$, where $A(G)$ is the adjacency matrix of $G$. Let $mu_1,cdots,mu_n$ be eigenvalues of $L_omega(G)$. Then the Laplacian energy of $G$ with respect to $omega$ defined as $LE_omega (G)=sum_{i=1}^nbig|mu_i - overline{omega}big|$, where $overline{omega}$ is the average of $omega$, i.e., $overline{omega}=dfrac{sum_{i=1}^{n}omega(v_i)}{n}$. In this paper we consider several natural vertex weights of $G$ and obtain some inequalities between the ordinary and Laplacian energies of $G$ with corresponding vertex weights. Finally, we apply our results to the molecular graph of toroidal fullerenes (or achiral polyhex nanotorus).[5mm] noindenttextbf{Key words:} Energy of graph, Laplacian energy, Vertex weight, Topological index, toroidal fullerenes.