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Let be a simple connected graph containing a perfect matching. For a positive integer , , is said to be bipartite matching extendable if every bipartite matching of with is included in a perfect matching of . In this paper, we show that bipartite matching extendability of some special graphs.

Let R=Z♭1♭2♭3×Zq2 be a commutative ring, where ♭1,♭2,♭3 are distinct primes, and q is any prime integer. A zero divisor graph JR of ring R is a graph with vertex set consist of zero divisors elements of R and any two vertices a,b are adjacent if and only if ab=0. A topological index is a numerical number associated with the graph and may be helpful to correlate the graph with certain of its physical/chemical properties. In this paper, we have computed some eccentricity based topological indices of JR, namely, atom-bond connectivity index (ABC5), eccentricity-based harmonic index of fourth type (H4J), geometric-arithmetic eccentricity index (GA4J), eccentricity-based third Zagreb index, and eccentricity-based first Zagreb index.

Topological indices (TIs) assign a numeric value to a graph or a molecular structure. Due to their ability to predict the physiochemical properties of a molecular graph, several TIs have been introduced and studied, mainly based on degree and distance. For a vertex v, the maximum distance of v from any other vertex in a graph G is called the eccentricity of v, which is denoted by σv, in G. The eccentricity of vertices gained special attention among the distance-based or degree-distance based TIs. An important TI in the class of eccentricity-dependent TIs is an eccentricity-entropy index. Furthermore, other eccentricity-dependent TIs such as eccentric-connectivity index, total-eccentricity index, and the first Zagreb index have also been extensively studied. On the other hand, dendrimers came out as unique polymeric macromolecules because of extensively branched three-dimensional architectural characteristics. This structure design prepares for various unique properties of dendrimers, including monodispersity, multivalency, uniform size, globular shape, water solubility with hydrophobic internal cavities, and a high degree of branching. These properties make them attractive candidates for different applications. PAMAM (polyamidoamine) dendrimers are promising polymers that can be successfully used in various biomedical applications. The PAMAM dendrimers having different structures such as a primary amine as the end group or porphyrin core have been studied through graph-theoretic parameters. This paper studies these two types of PAMAM dendrimers through eccentricity-dependent parameters. In particular, we establish formulae of eccentricity entropy for two types of PAMAM dendrimers. Moreover, we also derive analytic formulae of some other significant TIs from the class of eccentricity-dependent TIs. Furthermore, we apply graphical tools to demonstrate the trends of the values in the obtained results.