The application of graph theory in chemical and molecular structure research far exceeds people’s expectations, and it has recently grown exponentially. In the molecular graph, atoms are represented by vertices and bonded by edges. In this report, we study the M-polynomial of line graph of \(HAC_{5}C_{6}C_{7}[p,q]\) and recover many degree-based topological indices from it.

In this paper we aim to compute some Zagreb type polynomials of Möbius Ladder. Moreover we compute redefined Zagreb indices of Möbius Ladder.

The application of graph theory in chemical and molecular structure research far exceeds people’s expectations, and it has recently grown exponentially. In the molecular graph, atoms are represented by vertices and bonded by edges. In this report, we study the Zagreb-polynomials of line graph of \(HAC_{5}C_{6}C_{7}[p,q]\) and compute some degree-based topological indices from it.

Let \(k>0\) an integer. \(F\), \(\tau \), \(N\), \(N_{k}\), \(N_{k}^{(2)}\) and \(A\) denote the classes of finite, torsion, nilpotent, nilpotent of class at most \(k\), group which every two generator subgroup is \(N_{k}\) and abelian groups respectively. The main results of this paper is, firstly, we prove that, in the class of finitely generated \(\tau N\)-group (respectively \(FN\)-group) a \((FC)\tau \)-group (respectively \((FC)F\)-group) is a \(\tau A\)-group (respectively is \(FA\)-group). Secondly, we prove that a finitely generated \(\tau N\)-group (respectively \(FN\)-group) in the class \(((\tau N_{k})\tau ,\infty)\) (respectively \(((FN_{k})F,\infty)\)) is a \(\tau N_{k}^{(2)}\)-group (respectively \(FN_{k}^{(2)}\)-group). Thirdly we prove that a finitely generated \(\tau N\)-group ( respectively \(FN\)-group) in the class \(((\tau N_{k})\tau ,\infty)^{\ast}\) (respectively \(((FN_{k})F,\infty)^{\ast}\)) is a \(\tau N_{c}\)-group (respectively \(FN_{c}\)-group) for certain integer \(c\) and we extend this results to the class of \(NF\)-groups.

The generalized hierarchical product of graphs was introduced by L. Barrière et al. in 2009. In this paper, reformulated first Zagreb index of generalized hierarchical product of two connected graphs and hence as a special case cluster product of graphs are obtained. Finally using the derived results the reformulated first Zagreb index of some chemically important graphs such as square comb lattice, hexagonal chain, molecular graph of truncated cube, dimer fullerene, zig-zag polyhex nanotube and dicentric dendrimers are computed.

The atom-bond connectivity (ABC) index of a graph \(G=(V,E)\) is defined as \(ABC(G)=\sum_{v _{i}v_{j} \in E}\sqrt{(d_{i}+d_{j}-2)/(d_{i}d_{j})}\), where \(d_{i}\) denotes the degree of vertex \(v_{i}\) of \(G\). Due to its interesting applications in chemistry, this molecular structure descriptor has become one of the most actively studied vertex-degree-based graph invariants. Many efforts were made towards the elementary problem of characterizing tree(s) with minimal ABC index, which remains open and was coined as the ABC index conundrum”. Up to date, quite a few significant results have been obtained. In the course of research computer search plays a non-negligible role. In the present paper we review the state of the art of the problem. In addition we intend to demonstrate that, repeating the procedure “searching – conjecturing – proving” can be an applicable paradigm to cope with elusive problems of extremal graph characterization.

Let \(G\) be a simple, finite and connected graph. A graph is said to be decomposed into subgraphs \(H_1\) and \(H_2\) which is denoted by \(G= H_1 \oplus H_2\), if \(G\) is the edge disjoint union of \(H_1\) and \(H_2\). Assume that \(G= H_1 \oplus H_2 \oplus \cdots \oplus H_k\) and if each (H_i\), \(1 \leq i \leq k\), is a path or cycle in \(G\), then the collection of edge-disjoint subgraphs of \(G\) denoted by \(\psi\) is called a path decomposition of \(G\). If each \(H_i\) is a path in \(G\) then \(\psi\) is called an acyclic path decomposition of \(G\). The minimum cardinality of a path decomposition of \(G\), denoted by \(\pi (G)\), is called the path decomposition number and the minimum cardinality of an acyclic path decomposition of \(G\), denoted by \(\pi_a(G)\), is called the acyclic path decomposition number of \(G\). In this paper, we determine path decomposition number for a number of graphs in particular, the Cartesian product of graphs. We also provided bounds for \(\pi(G)\) and \(\pi_a(G)\) for these graphs.

The objective of the present study was to investigate and compare the efficacy, physiochemical equivalence and purity of API’s of different brands of cetirizine 2HCl tablets (10mg) with that of multinational brands available in local market of Lahore, Pakistan. The present work also provides awareness to allergy patients about most effective local brands with reasonable prices, without any health risk. Spectrophotometric method was used for chemical assay, TLC technique was subjected for the purity of API’s and physiochemical parameters was employed i.e; weight variation, hardness, friability, thickness, disintegration time and dissolution study as well as price fluctuation in PK rupees. The results of all physiochemical parameters and chemical assay were found to be within limit and meet to the pharmacopeial standards. TLC technique was developed to check the quality and purity of API, all the active ingredients showed the similar \(R_{f}\)-values without any impurity (0.38-0.39). From the test results and plots, it was observed and concluded that an economical and quality products can be prescribed for allergy patients whether they are manufactured by the local or multinational pharmaceutical companies without any health risk.

For an arborescence \(A_r\), a directed pathos total digraph \(Q=DPT(A_r)\) has vertex set \(V(Q)=V(A_r)\cup A(A_r)\cup P(A_r)\), = where \(V(A_r)\) is the vertex set, \(A(A_r)\) is the arc set, and \(P(A_r)\) is a directed pathos set of \(A_r\). The arc set \(A(Q)\) consists of the following arcs: \(ab\) such that \(a,b \in A(A_r)\) and the head of \(a\) coincides with the tail of \(b\); \(uv\) such that \(u,v \in V(A_r)\) and \(u\) is adjacent to \(v\); \(au\) \((ua)\) such that \(a\in A(A_r)\) and \(u \in V(A_r)\) and the head (tail) of \(a\) is \(u\); \(Pa\) such that \(a \in A(A_r)\) and \(P \in P(A_r)\) and the arc \(a\) lies on the directed path \(P\); \(P_iP_j\) such that \(P_i, P_j \in P(A_r)\) and it is possible to reach the head of \(P_j\) from the tail of \(P_i\) through a common vertex, but it is possible to reach the head of \(P_i\) from the tail of \(P_j\). For this class of digraphs we discuss the planarity; outerplanarity; maximal outerplanarity; minimally nonouterplanarity; and crossing number one properties of these digraphs. The problem of reconstructing an arborescence from its directed pathos total digraph is also presented.

In this paper, we use Riccati transformation technique to establish some new oscillation criteria for the second order nonlinear dynamic equation with damping on time scales $$(r(t)(x^\Delta(t))^\alpha)^\Delta-p(t)(x^\Delta(t))^\alpha+q(t)f(x(t))=0.$$ Our results not only generalize some existing results, but also can be applied to the oscillation problems that are not covered in literature. Finally, we give some examples to illustrate our main results.