Every year, various experiments emerge in which a strong link between topological chemical structures and their properties is found. These properties are numerous such as melting point, boiling point, and drug toxicity. Topological index is the functional tool to determine these properties. This research paper will analyze some of the molecular drug structures, i.e., hyaluronic acid-paclitaxel conjugates G n , anticancer drug SP n , polyomino chain of n -cycle Z n , triangular benzenoid T n , and circumcoronene benzenoid series H k using multicriteria decision-making techniques including TOPSIS and SAW. The topological indices used in this research paper include the Randić index for α = 1 , − 1 , 1 / 2 , the augmented Zagreb index and the forgotten topological index.

Chemical Graph entropy plays a significant role to measure the complexity of chemical structures. It has explicit chemical uses in chemistry, biology, and information sciences. A molecular structure of a compound consists of many atoms. Especially, the hydrocarbons is a chemical compound that consists of carbon and hydrogen atoms. In this article, we discussed the concept of subdivision of chemical graphs and their corresponding line chemical graphs. More preciously, we discuss the properties of chemical graph entropies and then constructed the chemical structures namely triangular benzenoid, hexagonal parallelogram, and zigzag edge coronoid fused with starphene. Also, we estimated the degree-based entropies with the help of line graphs of the subdivision of above mentioned chemical graphs.

We consider the problem of extending partial edge colorings of cartesian products of graphs. More specifically, we suggest the following Evans-type conjecture: If G is a graph where every precoloring of at most k precolored edges can be extended to a proper χ ′ (G)-edge coloring, then every precoloring of at most k + 1 edges of G K 2 is extendable to a proper (χ ′ (G)+1)-edge coloring of G K 2 . In this paper we verify that this conjecture holds for trees, complete and complete bipartite graphs, as well as for graphs with small maximum degree. We also prove versions of the conjecture for general regular graphs where the precolored edges are required to be independent.

Given a set of k colors and a graph G with a subset S of precolored edges (a partial k-edge coloring of G), we consider the problem of determining whether G has a proper edge coloring of G with the same k colors (an extension of the partial coloring) where the colors of edges in S are not changed. If such a coloring exists, then the partial k-coloring is called extendable.Some scheduling problems as well as some combinatorial problems can be reformulated as partial edge coloring extension problems for corresponding graphs. Partial edge coloring extension problems seem to have been first considered in connection with the problem of completing partial Latin squares. In 1960 Evans stated his conjecture that any partial Latin square of size n with at most n − 1 non-empty cells can be completed to a Latin square of size n. In terms of edge colorings this is equivalent to the statement that any proper partial n-edge coloring of the balanced complete bipartite graph K n,n with at most n − 1 precolored edges is extendable. This classical conjecture was proved by Smetaniuk (1981), and also independently by Andersen and Hilton (1983). Moreover, Andersen and Hilton completely characterized which partial Latin squares of size n with n non-empty cells that cannot be completed to a Latin square of size n. In addition, Andersen (1985) characterized partial Latin squares of size n with n+1 non-empty cells that are completable to Latin squares of size n.More recently, the problem of extending a partial edge coloring where the precolored edges form a matching has been considered by Edwards et al. (2018). Casselgren, Markstrom and Pham (2020) studied questions on extending partial edge colorings of the n-dimensional hypercubes Q n . In particular, they obtained an analogue of the positive solution to Evans' conjecture on completing partial Latin squares by proving that every proper partial edge coloring of at most n − 1 edges of Q n can be extended to a proper n-edge coloring of Q n . They also characterized which partial edge colorings of Q n with precisely n precolored edges are extendable to proper n-edge colorings of Q n .In this thesis we study similar partial edge coloring extension problems for trees. Let T be a tree with maximum degree ∆(T ). First, we characterize which partial edge colorings with at most ∆(T ) precolored edges in T that are extendable to proper ∆(T )-edge colorings, thereby proving an analogue of the aforementioned result by Andersen and Hilton for Latin squares. Then, we prove an analogue for trees of the result of Andersen by characterizing exactly which precolorings of at most ∆(T ) + 1 precolored edges in a tree T that are extendable to ∆(T )-edge colorings of T . We also prove sharp conditions on when it is possible to extend a precolored matching or a collection of connected precolored subgraphs of a tree T to a ∆(T )-edge coloring of T . Finally, we consider the problem of avoiding a given (not necessarily proper) partial edge coloring.