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Nina Zubrilina

R eferences [1] B. Bollobás, D. Mitsche and P. Pralat, Metric dimension for random graphs , (2012). arXiv:1208.3801 [2] G. Chartrand, C. Poisson and P. Zhang, Resolvability and the upper dimension of graphs , Comput. Math. Appl. 39 (2000) 19–28. doi:10.1016/S0898-1221(00)00126-7 [3] G. Chartrand, L. Eroh, M.A. Johnson and O.R. Oellermann, Resolvability in graphs and the metric dimension of a graph , Discrete Appl. Math. 105 (2000) 99–113. doi:10.1016/S0166-218X(00)00198-0 [4] F. Harary and R.A. Melter, On the metric dimension of a

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Sunilkumar M. Hosamani

polarity [ 14 ]. In a comprehensive study of numerous properties of octane isomers, Randić et. al [ 10 , 12 , 14 ] have used single molecular descriptors and concluded that different physicochemical properties depend on different descriptors. So far in the literature of chemical graph theory, domination parameters have not been used to predict the physical properties of chemical compounds. Therefore, in the present study an attempt has been made to study physical properties of octane isomers by using domination parameters. The values of γ c , γ t and γ t

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Manoj Changat, Ferdoos Hossein Nezhad, Henry Martyn Mulder and N. Narayanan

(2002) 349–370. doi:10.1016/S0012-365X(01)00296-5 [10] H.M. Mulder, The Interval Function of a Graph (MC Tracts 132, Mathematisch Centrum, Amsterdam, 1980). [11] H.M. Mulder, Transit functions on graphs ( and posets ), in: Convexity in Discrete Structures (M. Changat, S. Klavžar, H.M. Mulder, A. Vijayakumar, Eds.), Lecture Notes Ser. 5, Ramanujan Math. Soc. (2008) 117–130. [12] H.M. Mulder and L. Nebeský, Axiomatic characterization of the interval function of a graph , European J. Combin. 30 (2009) 1172–1185. doi:10.1016/j.ejc.2008

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Xueliang Li, Yaping Mao and Ivan Gutman

.L. Puertas, Steiner distance and convexity in graphs , European J. Combin. 29 (2008) 726–736. doi:10.1016/j.ejc.2007.03.007 [6] G. Chartrand, O.R. Oellermann, S.L. Tian and H.B. Zou, Steiner distance in graphs , Časopis Pest. Mat. 114 (1989) 399–410. [7] L. Chen, X. Li and M. Liu, The ( revised ) Szeged index and the Wiener index of a nonbipartite graph , European J. Combin. 36 (2014) 237–246. doi:10.1016/j.ejc.2013.07.019 [8] P. Dankelmann, O.R. Oellermann and H.C. Swart, The average Steiner distance of a graph , J. Graph Theory 22 (1996

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B. Basavanagoud, Veena R. Desai and Shreekant Patil

}\left( {{K_n}} \right) = \frac{{n\left( {n - 2} \right) \cdot {{(n - 1)}^{3\alpha + 1}}}}{2}.$ Lemma 7 (c.f. [ 3 ]). Let G be a graph with n vertices and m edges. Then M 1 ( G ) ≤ m ( 2 m n − 1 + n − 2 ) . $$\begin{array}{} \displaystyle {M_1}\left( G \right) \le m\left( {\frac{{2m}}{{n - 1}} + n - 2} \right). \end{array}$$ (4) Lemma 8 (c.f. [ 4 ]). Let G be a graph with n vertices and m edges, m > 0. Then the equality M 1 ( G ) = m ( 2 m n − 1 + n − 2 ) $$\begin{array}{} \displaystyle {M_1}\left( G \right) = m\left( {\frac{{2m}}{{n - 1}} + n

Open access

V. Lokesha, T. Deepika, P. S. Ranjini and I. N. Cangul

these topological indices. Muhammad Faisal Nadeem et. al., [ 17 ], computed generalized Randic, general Zagreb, general sum-connectivity, ABC , GA , ABC 4 , and GA 5 indices of the line graphs of 2D-lattice, nanotube and nanotorus of TUC 4 C 8 [ p , q ] by using the concept of subdivision. Sunil Hosamani [ 12 ], worked on computing sanskruti index of certain nanostructures. Computed the expressions for the Sanskruti index of the line graph of subdivision graph of the 2D-lattice, nanotube and nanotorus of TUC 4 C 8 [ p , q ]. Recently, C. K. Gupta

Open access

Ruijuan Li and Bin Sheng

Combin. 32 (2016) 1805–1816. doi:10.1007/s00373-015-1672-9 [5] N. Dean and B.J. Latka, Squaring the tournament—an open problem , Congr. Numer. 109 (1995) 73–80. [6] D. Fidler and R. Yuster, Remarks on the second neighborhood problem , J. Graph Theory 55 (2007) 208–220. doi:10.1002/jgt.v55:3 [7] D.C. Fisher, Squaring a tournament: a proof of Dean’s conjecture , J. Graph Theory 23 (1996) 43–48. doi:10.1002/(SICI)1097-0118(199609)23:1⟨43::AID-JGT4⟩3.0.CO;2-K [8] S. Ghazal, Seymour’s second neighbourhood conjecture for tournaments

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Cihat Abdioğlu, Ece Yetkin Çelikel and Angsuman Das

Abstract

In this paper we initiate the study of Armendariz graph of a commutative ring R and investigate the basic properties of this graph such as diameter, girth, domination number, etc. The Armendariz graph of a ring R, denoted by A(R), is an undirected graph with nonzero zero-divisors of R[x] (i.e., Z(R[x])*) as the vertex set, and two distinct vertices f(x)=i=0naixi and g(x)=j=0mbjxj are adjacent if and only if aibj = 0, for all i, j. It is shown that A(R), a subgraph of Γ(R[x]), the zero divisor graph of the polynomial ring R[x], have many graph properties in common with Γ(R[x]).

Open access

Jing Luo, Zhongxun Zhu and Runze Wan

R eferences [1] B. Bollobás, Modern Graph Theory (Springer-Verlag, 1998). doi:10.1007/978-1-4612-0619-4 [2] J. Guo, On the second largest Laplacian eigenvalue of trees , Linear Algebra Appl. 404 (2005) 251–261. doi:10.1016/j.laa.2005.02.031 [3] C.-X. He and H.-Y. Shan, On the Laplacian coefficients of bicyclic graphs , Discrete Math. 310 (2010) 3404–3412. doi:10.1016/j.disc.2010.08.012 [4] A. Ilić, Trees with minimal Laplacian coefficients , Comput. Math. Appl. 59 (2010) 2776–2783. doi:10.1016/j.camwa.2010.01.047 [5] S. Li

Open access

Brahim Benmedjdoub, Isma Bouchemakh and Éric Sopena

integer distance graphs , Discrete Math. 191 (1998) 113–123. doi:10.1016/S0012-365X(98)00099-5 [8] F. Kramer and H. Kramer, Un probleme de coloration des sommets d’un graphe , C. R. Acad. Sci. Paris A 268 (1969) 46–48. [9] F. Kramer and H. Kramer, A survey on the distance-colouring of graphs , Discrete Math. 308 (2008) 422–426. doi:10.1016/j.disc.2006.11.059 [10] K.-W. Lih and W.-F. Wang, Coloring the square of an outerplanar graph , Taiwanese J. Math. 10 (2006) 1015–1023. doi:10.11650/twjm/1500403890 [11] D.D.-F. Liu, From rainbow to