Characterizing Trees with Minimal ABC Index with Computer Search: A Short Survey

ODAM-Vol. 1 (2018), Issue 2, pp. 01–09 | Open Access Full-Text PDF
Yiming Zheng, Wenshui Lin, Qi’an Chen, Linshan Huang, Zhixi Wu
Abstract: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.
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Path Decomposition Number of Certain Graphs

ODAM-Vol. 1 (2018), Issue 1, pp. 26–32 | Open Access Full-Text PDF
Opeyemi Oyewumi, Abolape Deborah Akwu,Theresa Iveren Azer
Abstract: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.
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In Vitro Comparison of Different Brands of Cetirizine 2HCl Tablets by Spectroscopy and Thin Layer Chromatography (TLC) in Local Market of Lahore, Pakistan

EASL-Vol. 1 (2018), Issue 2, pp. 01–09 | Open Access Full-Text PDF
Sajid Mahmood, Ammara Rafiq, Zaheer Ahmad, Muhammad Ajmal
Abstract: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.
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Directed Pathos Total Digraph of an Arborescence

EASL-Vol. 1 (2018), Issue 1, pp. 29–42 | Open Access Full-Text PDF
M. C. Mahesh Kumar, H. M. Nagesh
Abstract: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.
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Oscillation behavior of second order nonlinear dynamic equation with damping on time scales

OMA-Vol. 2 (2018), Issue 2, pp. 78–88 | Open Access Full-Text PDF
Fanfan Li, Zhenlai Han
Abstract: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.
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Different Approaches for the Synthesis of Zinc Oxide Nanoparticles

OJC-Vol. 1 (2018), Issue 1, pp. 19–25 | Open Access Full-Text PDF
Zaheer Ahmad, Farman Ullah Khan, Sajid Mahmood, Tariq Mahmood, Aisha Shamim
Abstract:In this work we have described the synthesis of Zinc Oxide nanoparticles through chemical and biological methods. For biological synthesis Aspargillus niger was used. The product obtained was characterized through different analytical techniques like XRD, SEM and EDX. The obtained results were matched with the literature. It was confirmed that the Zinc Oxide nanoparticles can also be prepared from Aspargillus niger.Which may be more ecofriendly and economical compared to other commonly used methods.
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Zero-Sum Flow Number of Some Grid Graphs

ODAM-Vol. 1 (2018), Issue 1, pp. 16–25 | Open Access Full-Text PDF
Muhammad Kamran Siddiqui, Muhammad Naeem, Muhammad Imran
Abstract:For an undirected graph \(G\), a zero-sum flow is an assignment of nonzero integers to the edges such that the sum of the values of all edges incident with each vertex is zero, and we call it a zero-sum \(k\)-flow if the absolute values of edges are less than \(k\). We define the zero-sum flow number of \(G\) as the least integer \(k\) for which \(G\) admitting a zero sum \(k\)-flow. In this paper we gave complete zero-sum flow and zero sum number for octagonal grid, generalized prism and book graph.
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Distance-based Indices Computation of Symmetry Molecular Structures

OMS-Vol. 2 (2018), Issue 1, pp. 323–337 Open Access Full-Text PDF
Li Yan, Mohammad Reza Farahani, Wei Gao
Abstract:Most of molecular structures have symmetrical characteristics. It inspires us to calculate the topological indices by means of group theory. In this paper, we present the formulations for computing the several distance-based topological indices using group theory. We solve some examples as applications of our results.
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On Graph Invariants of Oxide Network

EASL-Vol. 1 (2018), Issue 1, pp. 23–28 | Open Access Full-Text PDF
Muhammad Imran, Asima Asghar, Abdul Qudair Baig
Abstract: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 several Zagreb polynomials and Redefined Zagreb indices of Oxide Network.
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