International Journal of Pure and Applied Mathematics Volume 6 No. 4 07, 035-04 ISSN: 3-8080 printed ersion); ISSN: 34-3395 on-line ersion) url: http://www.ijpam.eu doi: 0.73/ijpam.6i4.8 PAijpam.eu ON TOPOLOGICAL INDICES FOR THE LINE GRAPH OF FIRECRACKER GRAPH Ashaq Ali, Hifza Iqbal, Waqas Nazeer 3, Shin Min Kang 4, Department of Mathematics and Statistics The Uniersity of Lahore Lahore, 54000, PAKISTAN 3 Diision of Science and Technology Uniersity of Education Lahore, 54000, PAKISTAN 4 Department of Mathematics and RINS Gyeongsang National Uniersity Jinju, 588, KOREA Abstract: In this report, we compute newly defined topological indices, namely, AG index, SK index, SK index and SK index for the line graph of Firecracker graph. We also compute sum connectiity index and modified Randić index of underling graph. AMS Subject Classification: 05C90 Key Words: line graph, topological index, Firecracker graph. Introduction Cheminformatics is an emerging field in which quantitatie structure-actiity and Structure-property relationships predict the biological actiities and properties of nano-material see [, 3, 3, 5]. In these studies, some physcio-chemical properties and topological indices are used to predict bioactiity of the chemical compounds see [4, 5, 4]. Receied: 07-07-4 Reised: 07-0-3 Published: Noember 3, 07 c 07 Academic Publications, Ltd. url: www.acadpubl.eu Correspondence author
036 A. Ali, H. Iqbal, W. Nazeer, S.M. Kang The branch of chemistry which deals with the chemical structures with the help of mathematical tools is called mathematical chemistry. Chemical graph theory is that branch of mathematical chemistry which applies graph theory to mathematical modeling of chemical phenomena. In chemical graph theory a molecular graph is a simple graphhaing no loops and multiple edges) in which atoms and chemical bonds between them are represented by ertices and edges respectiely. A graph with ertex set VG) and edge set EG) is connected, if there exist a connection between any pair of ertices in G. The degree of a ertex is the number of ertices which are connected to that fixed ertex by the edges. The number of ertices of G, adjacent to a gien ertex, is the degree of this ertex, and will be denoted by d. The concept of degree in graph theory is closely related to the concept of alence in chemistry. One can see [5] for details on basics of graph theory. The topological index of a molecule structure can be considered as a nonempirical numerical quantity which quantifies the molecular structure and its branching pattern in many ways. In this point of iew, the topological index can be regarded as a score function which maps each molecular structure to a real number and is used as a descriptor of the molecule under testing. Topological indices gies a good predictions of ariety of physico-chemical properties of chemical compounds containing boiling point, heat of eaporation, heat of formation, chromatographic retention times, surface tension, apor pressure and partition coefficients could be rationalized by the assumption that Wiener index is roughly proportional to the an der Waals surface area of the respectie molecule. For details about topological indices, recently, we refer to [, 8, 9, 0, ]. The first Zagreb index and the second Zagreb index, introduced by Gutman and Trinajstić [7] and are defined as M G) V G) Sum connectiity index is defined as d ) and M G) SCG) and modified Randić index is defined as mrg) max{d u,d }..
ON TOPOLOGICAL INDICES FOR... 037 Shigehalli and Kanabur [] introduced following new degree-based topological indices; AG G) SK G), SKG), SK G) u EG), du d. The line graph LG) of a graph G is the graph, each of whose ertex represents an edge of G and two of its ertices are adjacent if their corresponding edges are adjacent in G. In this article, we compute some newly defined degreebased topological indices of the line graph of Firecracker graph, shown in Figure. The Firecracker graph Fn,k) is the graph obtained by the concatenation of nk-stars by linking one leaf from each. The Fn,k) has order nk and size nk. F4,7) is shown in the Figure. Figure. The Firecracker graph F4,7) Figure. The line graph of Firecracker graph F4,7) The following lemmas [6] are useful in our main results. Lemma.. Let G be a graph with u, EG) and e = u EG). Then de =. Lemma.. Let G be a graph of order p and size q. Then the line graph LG) of G is a graph of order p and u, VG) size M G) q.
038 A. Ali, H. Iqbal, W. Nazeer, S.M. Kang. Computational Results In this section, we gie our computational results. Theorem.. Let G is the line graph of Firecracker graph and n k. Then ) SCG) = 7 k n 5kn6n k 4 3k ) mrg) = nk n 4 4. n 4 k n )k ) k. n 6 4k k 4 k 3 Proof. The graph G for n = 4 and k = 7 is shown in Figure. By using Lemma., it is easy to see that the order of G is nk out of which ertices are of degree 3, ertices are of degree k, n 3 ertices are of degree 4, nk ) ertices are of degree k, and n ertices are of degree k. Therefore by using Lemma., G has size nk 3nk8n 8. We partition the size of G into edges of the type Edu,d), where u is an edge. In G, there are eight types of edges in based on degrees of end ertices of each edge. The first edge partition E G) contains edges u, where d u = 3 and d = 4. Thesecond edge partition E G) contains edges u, whered u = 3 and d = k. The third edge partition E 3 G) contains edges u, where d u = 3 and d = k. The forth edge partition E 4 G) contains n 4 edges u, where d u = d = 4. The fifth edge partition E 5 G) contains n 6 edges u, where d u = 4 and d = k. The sixth edge partition E 6 G) contains k ) edges u, where d u = k and d = k. The seenth edge partition E 7 G) contains nk 5nk6n edges u, where d u = d = k and the eighth edge partition E 8 G) contains n )k ) edges u, where d u = k, d = k: SCG) u E G) u E 4 G) u E 7 G) u E G) u E 5 G) u E 8 G) u E 3 G) u E 6 G)
ON TOPOLOGICAL INDICES FOR... 039 = 7 3k k n 5kn6n k 4 mrg) u E G) u E 3 G) u E 5 G) u E 7 G) n 4 k n 6 4k = nk n 4 4. n )k ) k. max{d u,d } max{d u,d } u E G) max{d u,d } u E 4 G) max{d u,d } u E 6 G) max{d u,d } u E 8 G) Hence ) and ) hold. This completes the proof. k 4 k 3 max{d u,d } max{d u,d } max{d u,d } max{d u,d } Theorem.. Let G is the line graph of Firecracker graph and n k. Then ) AG G) = 7 3k) n 8 n 6)4k) 3 3k k 4)k 3) k k )k ) n )k )k ) nk )k 3). kk ) ) SKG) = 4 k 4n6nk k3 n 5nk. 3) SK G) = 7 k 4n 5nk k k4 n 4 7k3 n 4 nk. 4) SK G) = 634n k 7k3 n 5k 3nK 3nk k4 n. Proof. For the line graph of Firecracker graph G, when n k, we hae AG G)
040 A. Ali, H. Iqbal, W. Nazeer, S.M. Kang u E G) u E 4 G) u E 7 G) u E G) u E 5 G) u E 8 G) u E 3 G) u E 6 G) = 7 3k) n 8 n 6)4k) 3 3k k 4)k 3) k k )k ) n )k )k ) nk )k 3). kk ) SKG) u E G) u E 4 G) u E 7 G) u E G) u E 5 G) u E 8 G) = 4 k 4n6nk k3 n 5nk. SK G) u E G) u E 4 G) u E 7 G) u E G) u E 5 G) u E 8 G) u E 3 G) u E 3 G) u E 6 G) u E 6 G) = 7 k 4n 5nk k k4 n 4 7k3 n 4 nk.
ON TOPOLOGICAL INDICES FOR... 04 SK G) u E G) u E 3 G) u E 5 G) u E 7 G) ) du d u E G) du d u E 4 G) du d u E 6 G) du d = 634n k 7k3 n u E 8 G) Hence )-4) holds. This completes the proof. ) ) ) ) 5k 3nK 3nk k4 n. 3. Conclusions and Discussion In this article, we computed AG index, SK index, SK index, SK index, sum connectiity index and modified Randić index for the line graph of Firecracker graph. These results can play a ital role in determining properties of this network and its uses in industry, electronics, and pharmacy. References [] M. Ajmal, W. Nazeer, M. Munir, S.M. Kang, Y.C. Kwun, Some algebraic polynomials and topological indices of generalized prism and toroidal polyhex networks, Symmetry, 9 07), Article ID 5, pages. doi:0.3390/sym900005 [] E. Deutsch, S. Klaˇzar, S, M-polynomial and degree-based topological indices, Iranian J. Math. Chem., 6 05), 93-0. doi: 0.05/ijmc.05.006 [3] Editorial, The many faces of nanotech, Nature Nanotechnology, 007), 585. doi: 0.038/nnano.007.308 [4] I. Gutman, Molecular graphs with minimal and maximal Randić indices, Croat. Chem. Acta, 75 00), 357-369. [5] I. Gutman, Degree-based topological indices, Croat. Chem. Acta, 86 03), 35-36. doi: 0.556/cca94
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