Narumi-Katayama Index of Some Derived Graphs
Abstract
The Narumi-Katayama index of a graph G is equal to the productof degrees of all the vertices of G. In this paper, we examine the Narumi-Katayama index of some derived graphs such as a Mycielski graph, subdivision graphs, double graph, extended double cover graph, thorn graph, subdivision vertex join and edge join graphs.
References
[1] N. Alon, Eigenvalues and expanders, Combinatorica, 6(2)(1986), 83-96.
[2] M. Caramia and P. DellOlmo, A lower bound on the chromatic number of Mycielski graphs, Discrete Math., 235(1-2)(2001), 79-86.
[3] V. Chvatal, The minimality of the Mycielski graph, Lecture Notes in Math., 406 (1974), 243-246.
[4] K. L. Collins and K. Tysdal, Dependent edges in Mycielski graphs and 4-colorings of 4-skeletons, J. Graph Theory, 46(4)(2004), 285-296.
[5] K.C. Das, K. Xu and J. Nam, On Zagreb indices of graphs, Front. Math. China, 10(3)(2015), 567-582.
[6] N. De, S.M.A. Nayeem and A. Pal, Total eccentricity index of generalized hierarchical product of graphs, Intern. J. Appl. Computational Math., 1(3)(2015), 503-511.
[7] N. De, On eccentric connectivity index and polynomial of thorn graph, Appl. Math., 3(2012), 931-934.
[8] N. De, Augmented eccentric connectivity index of some thorn graphs, Intern. J. Appl. Math. Res., 1(4)(2012), 671-680.
[9] N. De, A. Pal and S.M.A. Nayeem, On the modified eccentric connectivity index of generalized thorn graph, Intern. J. Computational Math., Volume 2014 (2014), Article ID 436140,
8 pages.
[10] N. De, A. Pal and S.M.A. Nayeem, On some bounds and exact formulae for connective eccentric indices of graphs under some graph operations, Intern. J. Combinatorics, Volume
2014 (2014), Article ID 579257, 5 pages
[11] N. De, S.M.A. Nayeem and A. Pal, Connective eccentricity index of some thorny graphs, Ann. Pure Appl. Math., 7(1)(2014), 59-64.
[12] T. Dehghan-Zadeh, H. Hua, A.R. Ashrafi and N. Habibi, Remarks on a conjecture about Randific index and graph radius, Miskolc Math. Notes, 14(3)(2013), 845-850.
[13] M. Eliasi and B. Taeri, Four new sums of graphs and their Wiener indices, Discrete Appl. Math., 157(4)(2009), 794-803.
[14] M. Eliasi, A. Iranmanesh and I. Gutman, Multiplicative versions of rst Zagreb index, MATCH Commun. Math. Comput. Chem., 68(2012), 217-230.
[15] M. Eliasi, G. Raeisi and B. Taeri, Wiener index of some graph operations, Discret. Appl. Math., 160(9)(2012), 1333-1344.
[16] G.H. Fath-Tabar, Old and new Zagreb indices of graphs, MATCH Commun. Math. Comput. Chem., 65(2011), 79-84.
[17] D.C. Fisher, P.A. McKena and E.D. Boyer, Hamiltonicity, diameter, domination, packing, and biclique partitions of Mycielski's graphs, Discret. Appl. Math., 84(1-3)(1998), 93-105.
[18] I. Gutman and N. Trinajstic, Graph theory and molecular orbitals. Total φ-electron energy of alternant hydrocarbons, Chem. Phys. Lett., 17(4)(1972), 535-538.
[19] I. Gutman, Distance in thorny graph, Publications de I'Institut Mathematique (Beograd), 63(77)(1998), 31-36.
[20] I. Gutman, Multiplicative Zagreb Indices of Trees, Bull. Inter. Math. Virtual Inst., 1(1)(2011), 13-19.
[21] H. Hua, A.R. Ashrafi and L. Zhang, More on Zagreb coindices of graphs, Filomat, 26(6)(2012), 1215-1225.
[22] H. Hua, S. Zhang and K. Xu, Further results on the eccentric distance sum, Disc. Appl. Math., 160(1-2)(2012), 170-180.
[23] G. Indulal, Spectrum of two new joins of graphs and infinite families of integral graphs, [1] N. Alon, Eigenvalues and expanders, Combinatorica, 6(2)(1986), 83-96.
[24] M.H. Khalifeha, H. Yousefi-Azaria and A.R. Ashrafi, The first and second Zagreb indices of some graph operations, Disc. Appl. Math., 157(4)(2009), 804-811.
[25] J. Liu and Q. Zhang, Sharp Upper Bounds for Multiplicative Zagreb Indices, MATCH Commun. Math. Comput. Chem., 68(2012), 231-240.
[26] J. Mycielski, Sur le coloriage des graphes, Colloq. Math., 3(1955), 161-162.
[27] H. Narumi, M. Katayama, Simple topological index. A newly devised index characterizing the topological nature of structural isomers of saturated hydrocarbons, Mem. Fac. Engin.
Hokkaido Univ., 16(3)(1984), 209-214.
[28] T. Reti and I. Gutman, Relations between Ordinary and Multiplicative Zagreb Indices, Bull. Internat. Math. Virt. Inst., 2(2)(2012), 133-140.
[29] R. Todeschini, D. Ballabio and V. Consonni, Novel molecular descriptors based on functions of new vertex degrees. in: I. Gutman and B. Furtula (eds.), Novel Molecular Structure Descriptors - Theory and Applications I (pp. 73{100), Univ. Kragujevac, Kragujevac, 2010.
[30] R. Todeschini and V. Consonni, New local vertex invariants and molecular descriptors based on functions of the vertex degrees, MATCH Commun. Math. Comput. Chem., 64(2010),
359-372.
[31] Z. Tomovic and I. Gutman, Narumi-Katayama index of phenylenes, J. Serb. Chem. Soc., 66(4)(2001), 243-247.
[32] B. Zhou, Upper bounds for the Zagreb indices and the spectral radius of series-parallel graphs, Int. J. Quantum Chem., 107(4)(2007), 875-878.
[33] B. Zhou and I. Gutman, Further properties of Zagreb indices, MATCH Commun. Math. Comput. Chem., 54(2005), 233-239.
[34] K. Xu, K. Tang, H. Liu and J. Wang, The Zagreb indices of bipartite graphs with more edges, J. Appl. Math. & Informatics, 33(3-4)(2015), 365-377.
[35] K. Xu, H. Hua, A unified approach to extremal multiplicative Zagreb indices for trees, unicyclic and bicyclic graphs, MATCH Commun. Math. Comput. Chem., 68(2012), 241-
256.
[36] W. Yan, B.Y. Yang and Y.N. Yeh, The behavior of Wiener indices and polynomials of graphs under five graph decorations, Appl. Math. Lett., 20(3)(2007), 290-295.
[2] M. Caramia and P. DellOlmo, A lower bound on the chromatic number of Mycielski graphs, Discrete Math., 235(1-2)(2001), 79-86.
[3] V. Chvatal, The minimality of the Mycielski graph, Lecture Notes in Math., 406 (1974), 243-246.
[4] K. L. Collins and K. Tysdal, Dependent edges in Mycielski graphs and 4-colorings of 4-skeletons, J. Graph Theory, 46(4)(2004), 285-296.
[5] K.C. Das, K. Xu and J. Nam, On Zagreb indices of graphs, Front. Math. China, 10(3)(2015), 567-582.
[6] N. De, S.M.A. Nayeem and A. Pal, Total eccentricity index of generalized hierarchical product of graphs, Intern. J. Appl. Computational Math., 1(3)(2015), 503-511.
[7] N. De, On eccentric connectivity index and polynomial of thorn graph, Appl. Math., 3(2012), 931-934.
[8] N. De, Augmented eccentric connectivity index of some thorn graphs, Intern. J. Appl. Math. Res., 1(4)(2012), 671-680.
[9] N. De, A. Pal and S.M.A. Nayeem, On the modified eccentric connectivity index of generalized thorn graph, Intern. J. Computational Math., Volume 2014 (2014), Article ID 436140,
8 pages.
[10] N. De, A. Pal and S.M.A. Nayeem, On some bounds and exact formulae for connective eccentric indices of graphs under some graph operations, Intern. J. Combinatorics, Volume
2014 (2014), Article ID 579257, 5 pages
[11] N. De, S.M.A. Nayeem and A. Pal, Connective eccentricity index of some thorny graphs, Ann. Pure Appl. Math., 7(1)(2014), 59-64.
[12] T. Dehghan-Zadeh, H. Hua, A.R. Ashrafi and N. Habibi, Remarks on a conjecture about Randific index and graph radius, Miskolc Math. Notes, 14(3)(2013), 845-850.
[13] M. Eliasi and B. Taeri, Four new sums of graphs and their Wiener indices, Discrete Appl. Math., 157(4)(2009), 794-803.
[14] M. Eliasi, A. Iranmanesh and I. Gutman, Multiplicative versions of rst Zagreb index, MATCH Commun. Math. Comput. Chem., 68(2012), 217-230.
[15] M. Eliasi, G. Raeisi and B. Taeri, Wiener index of some graph operations, Discret. Appl. Math., 160(9)(2012), 1333-1344.
[16] G.H. Fath-Tabar, Old and new Zagreb indices of graphs, MATCH Commun. Math. Comput. Chem., 65(2011), 79-84.
[17] D.C. Fisher, P.A. McKena and E.D. Boyer, Hamiltonicity, diameter, domination, packing, and biclique partitions of Mycielski's graphs, Discret. Appl. Math., 84(1-3)(1998), 93-105.
[18] I. Gutman and N. Trinajstic, Graph theory and molecular orbitals. Total φ-electron energy of alternant hydrocarbons, Chem. Phys. Lett., 17(4)(1972), 535-538.
[19] I. Gutman, Distance in thorny graph, Publications de I'Institut Mathematique (Beograd), 63(77)(1998), 31-36.
[20] I. Gutman, Multiplicative Zagreb Indices of Trees, Bull. Inter. Math. Virtual Inst., 1(1)(2011), 13-19.
[21] H. Hua, A.R. Ashrafi and L. Zhang, More on Zagreb coindices of graphs, Filomat, 26(6)(2012), 1215-1225.
[22] H. Hua, S. Zhang and K. Xu, Further results on the eccentric distance sum, Disc. Appl. Math., 160(1-2)(2012), 170-180.
[23] G. Indulal, Spectrum of two new joins of graphs and infinite families of integral graphs, [1] N. Alon, Eigenvalues and expanders, Combinatorica, 6(2)(1986), 83-96.
[24] M.H. Khalifeha, H. Yousefi-Azaria and A.R. Ashrafi, The first and second Zagreb indices of some graph operations, Disc. Appl. Math., 157(4)(2009), 804-811.
[25] J. Liu and Q. Zhang, Sharp Upper Bounds for Multiplicative Zagreb Indices, MATCH Commun. Math. Comput. Chem., 68(2012), 231-240.
[26] J. Mycielski, Sur le coloriage des graphes, Colloq. Math., 3(1955), 161-162.
[27] H. Narumi, M. Katayama, Simple topological index. A newly devised index characterizing the topological nature of structural isomers of saturated hydrocarbons, Mem. Fac. Engin.
Hokkaido Univ., 16(3)(1984), 209-214.
[28] T. Reti and I. Gutman, Relations between Ordinary and Multiplicative Zagreb Indices, Bull. Internat. Math. Virt. Inst., 2(2)(2012), 133-140.
[29] R. Todeschini, D. Ballabio and V. Consonni, Novel molecular descriptors based on functions of new vertex degrees. in: I. Gutman and B. Furtula (eds.), Novel Molecular Structure Descriptors - Theory and Applications I (pp. 73{100), Univ. Kragujevac, Kragujevac, 2010.
[30] R. Todeschini and V. Consonni, New local vertex invariants and molecular descriptors based on functions of the vertex degrees, MATCH Commun. Math. Comput. Chem., 64(2010),
359-372.
[31] Z. Tomovic and I. Gutman, Narumi-Katayama index of phenylenes, J. Serb. Chem. Soc., 66(4)(2001), 243-247.
[32] B. Zhou, Upper bounds for the Zagreb indices and the spectral radius of series-parallel graphs, Int. J. Quantum Chem., 107(4)(2007), 875-878.
[33] B. Zhou and I. Gutman, Further properties of Zagreb indices, MATCH Commun. Math. Comput. Chem., 54(2005), 233-239.
[34] K. Xu, K. Tang, H. Liu and J. Wang, The Zagreb indices of bipartite graphs with more edges, J. Appl. Math. & Informatics, 33(3-4)(2015), 365-377.
[35] K. Xu, H. Hua, A unified approach to extremal multiplicative Zagreb indices for trees, unicyclic and bicyclic graphs, MATCH Commun. Math. Comput. Chem., 68(2012), 241-
256.
[36] W. Yan, B.Y. Yang and Y.N. Yeh, The behavior of Wiener indices and polynomials of graphs under five graph decorations, Appl. Math. Lett., 20(3)(2007), 290-295.
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2017-01-01
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