Iranian Journal of Mathematical Chemistry
Volume:12 Issue: 3, Summer 2021

The modified first Zagreb connection index ZC∗1 for a graph G is defined as ZC∗1 (G) = sum v∈V (G) dvτv , where dv is the degree of the vertex v and τv denotes the connection number of v (that is, the number of vertices at the distance 2 from the vertex v). Let Tn,α be the class of trees with order n and matching number α such that n > 2α−1. In this paper, we obtain the lower bounds on the modified first Zagreb connection index of trees belonging to the class Tn,α, for 2α − 1 < n < 3α + 2.

In this paper, we present two approaches to compute the resolvent energy of a digraph . The first method computes the energy by ER(G)=sum_{i=1})^nfrac{1}{n-Re(z_i)}, where Re(z_i )denotes the real part of the eigenvalue z_i of G. In the second method we define ER(G)=sum_{i=1}^nfrac{1}{n-σ_i}, where σ_i is the ith singular value of G. We prove some properties of resolvent energy for some special digraphs and determine the resolvent energy of unicyclic and bicyclic digraphs and present lower bound for resolvent energy of directed cycles.

Gutman et al. gave some relations for computing the Hosoya indices of two special benzenoid systems R_n and P_n. In this paper, we compute the Hosoya index and Merrifield-Simmons index of R_n and P_n by means of introducing four vectors for each benzenoid system and index. As a result, we compute the Hosoya index and the Merrifield-Simmons index of R_n and P_n by means of a product of a certain matrix of degree n and a certain vector.

In the present work, a Mathematical model is proposed for the control on the concentration of hydroxyl ion in the precursor solution to preserve low super saturation level, because in order to obtain the desired and high quality one dimensional Zinc oxide nanostructures it is important to control the super saturation of the reactants. It was observed that elevated super saturation amount support nucleation and moderate super saturation amount support crystal growth during the synthesis of one dimensional Zinc oxide nanostructures. The kinematic reactions in the precursor solution were observed with the help of Lengyel-Epistein theory. Experimentally, the synthesis of ZnO nanostructures was also performed through Aqueous chemical growth method.

The spectrum of arbitrary graph of finite order the exponential growth of the resolvent of graph G is one of the most investigated object during the last 50 years. In particular, the resolvent matrix is a matrix with property that all of its eigenvalues are outside the spectra of G. In this paper, we study the exponential growth of the resolvent of graph G. The exponential growth of resolvent energy of graph G was investigated.