Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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On Diameter, Cyclomatic Number and Inverse Degree of Chemical Graphs

  • Sharafdini, Reza (Department of Mathematics, Faculty of Science, Persian Gulf University) ;
  • Ghalavand, Ali (Department of Pure Mathematics, Faculty of Mathematical Science, University of Kashan) ;
  • Ashrafi, Ali Reza (Department of Pure Mathematics, Faculty of Mathematical Science, University of Kashan)
  • Received : 2019.06.20
  • Accepted : 2020.04.22
  • Published : 2020.09.30
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Abstract

Let G be a chemical graph with vertex set {v1, v1, …, vn} and degree sequence d(G) = (degG(v1), degG(v2), …, degG(vn)). The inverse degree, R(G) of G is defined as $R(G)={\sum{_{i=1}^{n}}}\;{\frac{1}{deg_G(v_i)}}$. The cyclomatic number of G is defined as γ = m - n + k, where m, n and k are the number of edges, vertices and components of G, respectively. In this paper, some upper bounds on the diameter of a chemical graph in terms of its inverse degree are given. We also obtain an ordering of connected chemical graphs with respect to the inverse degree.

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References

  1. X. Chen and S. Fujita, On diameter and inverse degree of chemical graphs, Appl. Anal. Discrete Math., 7(2013), 83-93. https://doi.org/10.2298/AADM121129022C
  2. K. C. Das, K. Xu and J. Wang, On inverse degree and topological indices of graphs, Filomat, 30(8)(2016), 2111-2120. https://doi.org/10.2298/FIL1608111D
  3. D. Dimitrov and A. Ali, On the extremal graphs with respect to variable sum exdeg index, Discrete Math. Lett., 1(2019), 42-48.
  4. R. Entringer, Bounds for the average distance-inverse degree product in trees, Combinatorics, Graph Theory, and Algorithms I, II, 335-352, New Issues Press, Kalamazoo, MI, 1999.
  5. P. Erdos, J. Pach and J. Spencer, On the mean distance between points of a graph, Congr. Numer., 64(1988), 121-124.
  6. S. Fajtlowicz, On conjectures of graffiti II, Congr. Numer., 60(1987), 189-197.
  7. A. Ghalavand and A. R. Ashra, Extremal graphs with respect to variable sum exdeg index via majorization, Appl. Math. Comput., 303(2017), 19-23. https://doi.org/10.1016/j.amc.2017.01.007
  8. A. Ghalavand, A. R. Ashrafi and I. Gutman, Extremal graphs for the second multiplicative Zagreb index, Bull. Int. Math. Virtual Inst., 8(2)(2018), 369-383.
  9. L. B. Kier and L. H. Hall, The nature of structure-activity relationships and their relation to molecular connectivity, European J. Med. Chem., 12(1977), 307-312.
  10. A. W. Marshall and I. Olkin, Inequalities: Theory of majorization and its applications, Mathematics in Science and Engineering 143, Academic Press, Inc. New York-London, 1979.
  11. K. Xu, Kexiang, and K. Ch. Das, Some extremal graphs with respect to inverse degree, Discrete Appl. Math., 203(2016), 171-183. https://doi.org/10.1016/j.dam.2015.09.004
  12. Z. Zhang, J. Zhang and X. Lu, The relation of matching with inverse degree of a graph, Discrete Math., 301(2005), 243-246. https://doi.org/10.1016/j.disc.2003.01.001