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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.

Measures of graph entropy have been widely used in information theory, biology, chemistry, and sociology, among other disciplines. The entropy of a probability distribution may be thought of as both an indicator of information and an indicator of uncertainty, and it is a number used in information theory to gauge the complexity of a graph. Mathematical and medicinal chemistry, including drug design, have previously made extensive use of information-theoretic network complexity measurements. Dendrimers belong to the world of molecular chemistry for stepwise controlled synthesis and to the world of polymers for repeating structures from monomers. A topological index is a numerical metric that describes the topology of a graph. Consequently , a topological index calculated for a molecular network is a quantitative indicator of molecular topology. Due to the simplicity of production and the speed at which these computations may be performed, these descriptors have gained great prominence in recent years. Characterizing molecular structure using topological methods, including numerical graph invariants, is a current trend in mathematics and computational chemistry. Such conceptual descriptors have also launched a broad range of applications in QSAR/QSPR investigations and is ideal for innovative molecular design, drug development, and risk assessment of chemicals. In this paper, various entropy measures along with the distance and degree based topological indices of highly efficient iridium cored electrophosphorescent dendrimer has been evaluated. Entropy measure is a topological feature that may be used to assess the complexity of chemical compounds.

Nanomaterials feature exceptional, one-of-a-kind qualities that might be used in electronics, medicine, and other industries. Due to their unique photonic, electrical, catalytic, and magnetic capabilities as well as their use in helping immunotherapy, drug carriers, biosensors, electronic devices, etc., two dimensional materials are becoming considerably more important. Two-dimensional nanomaterials called borophene have a variety of intriguing characteristics. In the fields of chemistry, material science, nanotechnology, and condensed matter physics, they left an indelible impression thanks to their astounding physical and molecular characteristics. The concept of modelling the structure of a molecule or chemical network to a chemical graph and then quantitatively analysing the resulting graph with the aid of topological descriptors was a major advance in the fields of mathematics and chemistry. Topological descriptors are used in a wide range of fields, including chemistry, information technology, biology, quantitative structure-property relationships (QSPR), quantitative structure-activity relationships (QSAR), and many more. New descriptors that more closely connect with the structural characteristics of a chemical structure are being developed by researchers working on this idea. Finding a chemical graph's structural descriptors may be done in a variety of ways. It is discovered that the M-polynomial is a very versatile and quick method for computing the degree-based descriptors of chemical graphs or networks. The degree-based descriptors of the $\beta_{12}$-Borophene nanosheet are established in this study utilising the M-polynomial technique. At the conclusion, the numerical and graphical comparison based on the identified analytic expressions is also provided.

For a connected graph G(V,E), let S ⊆ V(G) be a minimum strong geodetic set of G then a non-extreme vertex v ∈ S is said to be S-extreme vertex if v is not geodominated by any pair of vertices of G. A strong geodetic set S of vertices in a connected graph G is an strong open geodetic set if for each vertex v ∈ G, either v is a S-extreme vertex of G and v ∈ S or v is an internal vertex of an x — y geodesic for some x,y ∈ S. A strong open geodetic set of minimum cardinality is a minimum strong open geodetic set and this cardinality is the strong open geodetic number of a graph G, denoted by sog(G). In this paper we have discussed about some results related to open geodetic sets. Also, mainly we have the complexity property of strong open geodetic set problem for general graphs, chordal graphs, chordal bipartite graphs. Further few general results and some bounds are given to the strong open geodetic number of some graphs.

Let G = (V, E) be a graph. A vertex v is k–geodominated by a pair x,y ⊆ V(G), if v lies on an x – y geodesic of G of length exactly k. A set S ⊆ V(G) is a strong (k, r)geodetic set if each vertex v ∈ G/S lies on at least r distinct fixed k–geodesics between the pair of vertices in S. The minimum cardinality of the strong (k, r) geodetic set is a strong (k, r) geodetic number and is denoted by sg(k,r)(G). In this paper we prove that the strong (k, r) geodetic set is NP–complete for general graphs, bipartite graphs, chordal graphs and chordal bipartite graphs. Also we derived some results on strong (2,2) geodetic number.

For a graph G(V(G),E(G)), the set Z ⊆ V(G) is called a geodetic propagation set if Z is both a geodetic set as well as a propagating set. The cardinality of the minimum geodetic propagation set is called the geodetic propagation number. Few general results and also the computational complexity part of this concept for general,bipartite and chordal graphs are derived.

Geodetic number of a graph is one of the widely studied graph theoretic parameters concerning geodesic convexity in graphs. In this paper, we introduce a variant of this parameter namely, doubly geodetic number of a graph. For a connected graph G, a setSof vertices of Gis called a doubly geodetic set ofGif each vertex inV − Slies on atleast two distinct geodesics of vertices in S.The doubly geodetic numberd g G of G is the minimum cardinality of a doubly geodetic set. Any doubly geodetic set of cardinality d g G is called d g-set of G. In this paper, the doubly geodetic number of certain standard graphs is determined.

Broadcasting is one of the types of global communication in interconnection networks. Broadcasting from a vertex of a graph is the simplest communication problem. In this paper, we analyse the broadcasting for bloom graph and prove that the broadcast time of bloom graph is at most two time units more than the optimal time.

In this paper, we propose a new graph namely Sierpi_nski Gasket Rhombus SRn based on triangular graph like Sierpi_nski Gasket graph. Sierpi_nski Gasket Rhombus SRn is obtained by identifying two copies of Sierpi_nski Gasket graph along their side edges. Further we study some basic properties of Sierpi_nski Gasket Rhombus SRn and find hamiltonicity, pancyclicity and chromatic number of SRn.