Let G be a molecular graph. The total-eccentricity index of graph G is defined as the sum of eccentricities of all vertices of G. The extremal trees, unicyclic and bicyclic graphs, and extremal conjugated trees with respect to total-eccentricity index are known. In this paper, we extend these results and study the extremal conjugated unicyclic and bicyclic graphs with respect to total-eccentricity index.

Metal-Organic Networks (MON’s) is the central bone for the chemical compounds of the latest study for the energy department. The study of MON’s structure provides us numerous benefits in different fields related to chemical sciences, electrical and civil sectors. The MON’s structure is also used to restore different chemical compounds, especially those elements which can be used for the energy purpose such as hydrogen and carbon. Topological indices of the MON’s structure provide relationships between physical and chemical characteristics of the this compound such as melting points, boiling points, chemically stability, pressure, chemical reaction factors and many other basic properties. In this paper we calculate different topological indices based on first, second and third distances for two different metal-organic networks with expanding number of layers consisting on both organic ligands and metal vertices. A comparison between the calculated different kinds of the Topological Indices with the help of the numerical values and their graphical representation is also included.

Conjugated open ended cones in which the configured pentagons are consistent, lies in the circle of Fries Kekule structure [8]. This non-adjacent tightest configuration of pentagons as shown in the Fig. 1 is consistent with a Fries Kekule structure and thus provides the most stable cone. In the study, various topological indices of the same structure in regard of physico-concoction resources and bioactivity of substance mixtures are studied which further helps to study the behavior of chemical compounds. In this regard, Zagreb indices M_1^* (G) and M_2^* (G) of a molecular graph G are used to evaluate the complexity in chemical systems and biological organisms. In this manuscript, we consider two complex families of stable carbon nanocones and compute their ECI, TEI and eccentricity-based Zagreb indices.