Dendrimers are constructed by the successive addition of layers of branching groups. Topological indices are numerical numbers associated which are graph invariant up to isomorphism. In this report, we give closed form of M-polynomial of PETAA dendrimers and from this M-polynomial, we recover many topological indices. These topological indices help to predict physcio-chemical properties of the underling dendrimers, and help to the study of the properties of the materials of ship building. We also give graphical representation of our results.

Measurements of graphs and retrieving structural information of complex networks using degree-based network entropy have become an informational theoretical concept. This terminology is extended by the concept of Shannon entropy. In this paper, we introduce entropy with graphs having edge weights which are basically redefined Zagreb indices. Some bounds are calculated to idealize the performance in limiting different kinds of graph entropy. In addition, we discuss the structural complexity of connected graphs representing chemical structures. In this article, we have discussed the edge-weighted graph entropy with fixed number of vertices, with minimum and maximum degree of a vertex, regular graphs, complete graphs, complete bipartite graphs, and graphs associated with chemical structures.

Among topological descriptors topological indices are significant and they have a conspicuous role in chemistry. Dutch Windmill graph Dxy can be obtain by taking x copies of cycle Cy with a vertex in common. In this paper, we will compute some irregularity índices that are useful in quantitative structure activity relationship for Line Graph of Dutch Windmill graph.

Energy of a graph is defined as the sum of the absolute values of the eigenvalues of the adjacency matrix associated with the graph. In this research work, we find color energy, distance energy, Laplacian energy, and Seidel energy for the Dutch windmill graph of cycle lengths 4, 5, and 6. Also, we find the lower bounds of the double‐wheel graph for energy, Seidel energy, color energy, distance energy, Laplacian energy, and Harary energy.

Sheehalli and Kanabur presented new forms of topological indices known as SK indices which have many applications in chemical graph theory towards QSPR/QSAR. A simple connected graph Γ is called a cactus graph if and only if any two cycles in the graph have no common edge. In this paper, we will explore and determine the sharp upper bounds of cactus graph using different versions of SK indices. We will also characterize the extremal properties for these indices due to many applications in pharmacology and medicinal chemistry.

The graph entropy was proposed by Körner in the year 1973 when he was studying the problem of coding in information theory. The foundation of graph entropy is in information theory, but it was demonstrated to be firmly identified with some established and often examined graph-theoretic ideas. For instance, it gives an equal definition to a graph to be flawless, and it can likewise be connected to acquire lower bounds in graph covering problems. The objective of this study is to solve the open problem suggested by Kwun et al. in 2018. In this paper, we study the weighted graph entropy by taking augmented Zagreb edge weight and give bounds of it for regular, connected, bipartite, chemical, unicyclic, etc., graphs. Moreover, we compute the weighted graph entropy of certain nanotubes and plot our results to see dependence of weighted entropy on involved parameters.