A topological index (TI) is a real number that defines the relationship between a chemical structure and its properties and remains invariant under graph isomorphism. TIs defined for chemical structures are capable of predicting physical properties, chemical reactivity and biological activity. Several kinds of TIs have been defined and studied for different molecular structures. Graphene is the thinnest material known to man and is also extremely strong while being a good conductor of heat and electricity. With such unique features, graphene and its derivatives have found commercial uses and have also fascinated theoretical chemists. In this article, the neighbourhood sum degree-based M-polynomial and entropy measures have been computed for graphene, graphyne and graphdiyne structures. The proper analytical expressions for these indices are derived. The obtained results will enable theoretical chemists to study these exciting structures further from a structural perspective.

Background: Sodalite is a type of zeolite with an intricate structure comprising a system of interrelated cages and tunnels. It is extensively used in sieving applications due to its unique structure and properties. As a result, it finds several uses in water and air purifica-tion, radioactive decontamination, detergents, and so on. Due to the potential positive envi-ronmental impact of sodalite materials, analysing the molecule at a structural level becomes the need of the hour. Methods: Molecular descriptors form the basis of many convenient and cost-effective tech-niques for studying molecular structures. In this article, the neighbourhood sum-based de-scriptors are computed edge-partition techniques, simplifying the intricate structures with cages and tunnels into more simple graphs for mathematical convenience. Results: This article presents the calculated analytical expressions for molecular descriptors, specifically focusing on various neighbourhood sum degree-based indices in sodalite struc-tures. Conclusion: The results presented in this article establish the dependence of the physical properties of a molecule on its underlying structure using the computed molecular de-scriptors. The graphical comparison of the results provides a visual representation of the be-haviour of indices with respect to the molecular structure.

Dendrimers are well‐defined nanoparticles, which have far‐reaching application in the field of chemistry. Many efforts have been devoted to development of dendrimer due to thier unique structure and various properties and broad application. It helps in varieties of purposes as a catalyst in drug delivery and drug design. The topological descriptor analyze the structure–property relationship of chemical compounds. In this paper, some numerical expressions have been obtained to understand the behavioral pattern of chiral polyamidoamine (PAMAM) dendrimer and PAMAM anthracene moieties dendrimer. The analytical expression has been plotted and compared with varieties of indices to show how it varies between each indices.

A geodetic landmark set of a connected graph G(V, E) is a geodetic set S ⊆ (G) with the property that S is also a metric basis of G. The geodetic landmark number is the minimum cardinality of a geodetic landmark set in G.The geodetic landmark number of a graph G is denoted by gl(G). This paper mainly composed of two parts, one is the computational complexity and some results on geodetic landmark number and bounds. We have proved the geodetic landmark set problem is NP-complete for general graph. Some theorems are also proved regarding geodetic landmark. The geodetic landmark number of some simple graphs are calculated. Also, geodetic landmark number of interconnection networks like butterfly network, benes network and silicate graph of dimension r are calculated.

Let G be a graph with vertex set as V(G) and edge set as E(G) which is simple as well as connected. The problem of strong total monophonic set is to find the set of vertices T⊆V(G), which contains no isolated vertices, and all the vertices in V(G)\T lie on a fixed unique chordless path between the pair of vertices in T. The cardinality of strong total monophonic set which is minimum is defined as strong total monophonic number, denoted by sm_t(G). We proved the NP‐completeness of strong total monophonic set for general graphs. The strong total monophonic number of certain graphs and networks is derived. If l, m, n are positive integers with 5 ≤ l ≤ m ≤ n and m ≤ 2l − 1, then we can construct a connected graph G with strong monophonic number l and strong total monophonic number m.