Irregular topological indices of certain metal organic frameworks


 It is interesting to study the molecular topology that provides a base for relationship of physicochemical property of a definite molecule. The topology of a molecule and the irregularity of the structure plays a vital character in shaping properties of the structure like enthalpy and entropy. In this article, we are interested to calculate some irregular topological indices of two classes of metal organic frameworks (MOFs) namely BHT (Butylated hydroxytoluene) based metal (M = Co, Fe, Mn, Cr) organic frameworks (MBHT) and M1TPyP-M2 (TPyP = 5, 10, 15, 20-tetrakis (4-pyridyl) porphyrin and M1, M2, = Fe and Co) MOFs. Also we compare our results graphically.

organic frameworks (Kinoshita, 1959). Due to their design and synthesis, MOFs gain attention rapidly (Hoskins, 1989). Till now a large number of MOFs have been synthesized and used in many applications especially in gas catalysis (Hall et al., 2016;Lee et al., 2009;Roy et al., 2012), delivery of drugs (Horcajada et al., 2008;Mandal et al., 2017;Vallet-Regi et al., 2007), sensing (Sarkisov et al., 2012), separation (Kim et al., 2017;Li et al., 2009), storage (Kennedy et al., 2013;Murray et al., 2009;Rosi et al., 2003), and absorption (Czaja et al., 2009;Geier et al., 2013;Mu et al., 2010;Park et al., 2017;Queen et al., 2014). MOFs are capable of capturing industrial gases, such as CO 2 , SO 2 , NO, CO, NO 2 , etc. (Dietzel et al., 2009;Lee et al., 2015;Wu et al., 2009). These gases are very dangerous for our environment. For example, the CO 2 continuously changing our climate and effects greenhouse (Rodhe, 1990), acid rain, and smog is due to the emmision of SO 2 and NO 2 (Singh and Agarwal, 2007), and CO and NO are very harmful for humans (Olson and Phillips, 1997). For a healthy environment, it is necessary to control these dangerous gases. MOFs have ability to reduce the quantity of CO 2 at room temperature and low pressure. We can study the ability of MOFs to reduce flue gases in these articles (Chakarvarty et al., 2016;Howe et al., 2017;Tan et al., 2017;Yu et al., 2012). We can study for degree based topological invariants of metal-organic networks (Hong et al., 2020).
Let G(V,E) with vertex set V and edge set E be a connected graph of order n = |V(G)| and size m = |E(G)|. The number of edges associated with a vertex is the degree of that vertex. The quantitative topological categorization of irregularity of graphs has an increasing significance for analyzing the structure of deterministic and arbitrary networks and systems occurring in chemistry, biology and common networks. The idea of topological indices was given by Wiener (1947). He also found a strong relation among weiner index and physicochemical properties of compounds. But mathematicians did not work with interest on it for next 20 years. In the mid of 1970s, Wiener index gain popularity and gave some important research articles. After 1990s, a lot of work has done on other distance based topological indices closely related to Wiener index. Till now, thousands of toplogical indices (distance based and degree based topological indices) have found which plays a vital role in chemical graph theory. Now we present the irregularity topological indices that is calculated here. Albertson (1997)  Proof: We will use Figure 1 to prove all the above theorems. We can verify the values given in Table 1 for the edges of G1(c,d). 1.

Irregularity topological indices of 2D CoBHT(CO) lattice
Let G2(a,b) be the graph of 2D CoBHT(CO) lattice, where 'a' and 'b' are the unit cells in a row and column respectively. The molecular graph of G2(2,2) is shown below. We can verified that G2(a,b) has 27ab number of vertices and 36ab-2a-2b number of edges.

Theorem 2
Let G2(a,b) be the graph of 2D structure of CoBHT(CO) lattice, then the irregularity indices of G2(a,b) are:  Figure 2 to prove all above theorems. We can verify the values given in Table 2 for the edges of G2(a,b).

Graphical analysis and conclusions
Here we present the graphical analysis and comparison table (Table 3) of some of the irregularity indices of the graph of 2D structure of M1TPyP-M2 metal organic frameworks. Figure 3 contains the graphical values of Albertson index (AL(G)), irregularity IRL(G) and IRLU(G) indices, total irregularity index (IRRt), IRF(G) irregularity index, randic index (IRA(G)), irregularity index IRDIF(G), irregularity index IRLF(G), irregularity index LA(G), irregularity index IRDI(G), irregularity index IRGA(G), and irregularity index IRB(G). In this table, we can check the values of some irregularity indices for some different values of 'c' and 'd'. Now we present the graphical analysis and comparison table (Table 4) of some of the irregularity indices of the graph of 2D structure of CoBHT(CO) lattice. Figure 4 contains the graphical values of Albertson index (AL(G)), irregularity IRL(G) and IRLU(G) indices, total irregularity index IRRt(G), IRF(G) irregularity index, randic index IRA(G), irregularity index IRDIF(G), irregularity index IRLF(G), irregularity index LA(G), irregularity index IRDI(G), irregularity index IRGA(G), and irregularity index IRB(G). In Table 4 we can check the values of some irregularity indices for some different values of 'a' and 'b'.