Frontiers of Mathematics in China >
Zagreb indices of graphs
Received date: 06 Jan 2014
Accepted date: 18 Sep 2014
Published date: 01 Apr 2015
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The first Zagreb index M1(G) is equal to the sum of squares of the degrees of the vertices, and the second Zagreb index M2(G) is equal to the sum of the products of the degrees of pairs of adjacent vertices of the underlying molecular graph G. In this paper, we obtain lower and upper bounds on the first Zagreb index M1(G) of G in terms of the number of vertices (n), number of edges (m), maximum vertex degree (Δ), and minimum vertex degree (δ). Using this result, we find lower and upper bounds on M2(G). Also, we present lower and upper bounds on M2(G) +M2(G) in terms of n, m, Δ, and δ, where G denotes the complement of G. Moreover, we determine the bounds on first Zagreb coindex M1(G) and second Zagreb coindex M2(G). Finally, we give a relation between the first Zagreb index and the second Zagreb index of graph G.
Kinkar Ch. DAS , Kexiang XU , Junki NAM . Zagreb indices of graphs[J]. Frontiers of Mathematics in China, 2015 , 10(3) : 567 -582 . DOI: 10.1007/s11464-015-0431-9
1 |
Ashrafi A R, Došlić T, Hamzeh A. The Zagreb coindices of graph operations. Discrete Appl Math, 2010, 158: 1571-1578
|
2 |
Balaban A T, Motoc I, Bonchev D, Mekenyan O. Topological indices for structureactivity correlations. Topics Curr Chem, 1983, 114: 21-55
|
3 |
Biler P, Witkowski A. Problems in Mathematical Analysis. New York: Marcel Dekker, Inc, 1990
|
4 |
Bondy J A, Murty U S R. Graph Theory with Applications. New York: Macmillan Press, 1976
|
5 |
Bullen P S, Mitrinović D S, Vasić P M. Means and their inequalities. Dordrecht: Reidel, 1988
|
6 |
de Caen D. An upper bound on the sum of squares of degrees in a graph. Discrete Math, 1998, 185: 245-248
|
7 |
Dankelmann P, Hellwig A, Volkmann L. Inverse degree and edge-connectivity. Discrete Math, 2009, 309: 2943-2947
|
8 |
Dankelmann P, Swart H C, Van Den Berg P. Diameter and inverse degree. Discrete Math, 2008, 308: 670-673
|
9 |
Das K C. Sharp bounds for the sum of the squares of the degrees of a graph. Kragujevac J Math, 2003, 25: 31-49
|
10 |
Das K C. Maximizing the sum of the squares of the degrees of a graph. Discrete Math, 2004, 285: 57-66
|
11 |
Das K C. On comparing Zagreb indices of graphs. MATCH Commun Math Comput Chem, 2010, 63: 433-440
|
12 |
Das K C, Gutman I. Some properties of the second Zagreb index. MATCH Commun Math Comput Chem, 2004, 52: 103-112
|
13 |
Das K C, Gutman I, Horoldagva B. Comparison between Zagreb indices and Zagreb coindices. MATCH Commun Math Comput Chem, 2012, 68: 189-198
|
14 |
Das K C, Gutman I, Zhou B. New upper bounds on Zagreb indices. J Math Chem, 2009, 46: 514-521
|
15 |
Das K C, Xu K, Gutman I. On Zagreb and Harary indices. MATCH Commun Math Comput Chem, 2013, 70: 301-314
|
16 |
Eliasi M, Iranmanesh A, Gutman I. Multiplicative versions of first Zagreb index. MATCH Commun Math Comput Chem, 2012, 68: 217-230
|
17 |
Erdös P, Pach J, Spencer J. On the mean distance between points of a graph. Congr Numer, 1988, 64: 121-124
|
18 |
Fajtlowicz S. On conjectures of graffiti II. Congr Numer, 1987, 60: 189-197
|
19 |
Gutman I. Multiplicative Zagreb indices of trees. Bull Soc Math Banja Luka, 2011, 18: 17-23
|
20 |
Gutman I, Das K C. The first Zagreb index 30 years after. MATCH Commun Math Comput Chem, 2004, 50: 83-92
|
21 |
Gutman I, Ghorbani M. Some properties of the Narumi-Katayama index. Appl Math Lett, 2012, 25: 1435-1438
|
22 |
Gutman I, Ruščić B, Trinajstić N, Wilcox C F. Graph theory and molecular orbitals. XII. Acyclic polyenes. J Chem Phys, 1975, 62: 3399-3405
|
23 |
Gutman I, Trinajstić N. Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons. Chem Phys Lett, 1972, 17: 535-538
|
24 |
Mitrinović D S. Analytic Inequalities. Berlin-Heidelberg-New York: Springer, 1970
|
25 |
Narumi H, Katayama M. Simple topological index. A newly devised index characterizing the topological nature of structural isomers of saturated hydrocarbons. Mem Fac Engin Hokkaido Univ, 1984, 16: 209-214
|
26 |
Nikolić S, Kovačević G, Milićević A, Trinajstić N. The Zagreb indices 30 years after. Croat Chem Acta, 2003, 76: 113-124
|
27 |
Radon J. Über die absolut additiven Mengenfunktionen. Wiener Sitzungsber, 1913, 122: 1295-1438
|
28 |
Todeschini R, Consonni V. Handbook of Molecular Descriptors. Weinheim: Wiley-VCH, 2000
|
29 |
Todeschini R, Consonni V. New local vertex invariants and molecular descriptors based on functions of the vertex degrees. MATCH Commun Math Comput Chem, 2010, 64: 359-372
|
30 |
Xu K. The Zagreb indices of graphs with a given clique number. Appl Math Lett, 2011, 24: 1026-1030
|
31 |
Xu K, Das K C. Trees, unicyclic, and bicyclic graphs extremal with respect to multiplicative sum Zagreb index. MATCH Commun Math Comput Chem, 2012, 68: 257-272
|
32 |
Xu K, Das K C, Balachandran S. Maximizing the Zagreb indices of (n,m)-graphs. MATCH Commun Math Comput Chem, 2014, 72: 641-654
|
33 |
Xu K, Das K C, Tang K. On the multiplicative Zagreb coindex of graphs. Opuscula Math, 2013, 33: 197-210
|
34 |
Xu K, Hua H. A unified approach to extremal multiplicative Zagreb indices for trees, unicyclic and bicyclic graphs. MATCH Commun Math Comput Chem, 2012, 68: 241-256
|
35 |
Yan Z, Liu H, Liu H. Sharp bounds for the second Zagreb index of unicyclic graphs. J Math Chem, 2007, 42: 565-574
|
36 |
Zhang L, Wu B. The Nordhaus-Gaddum-type inequalities for some chemical indices. MATCH Commun Math Comput Chem, 2005, 54: 189-194
|
37 |
Zhou B. Upper bounds for the Zagreb indices and the spectral radius of series-parallel graphs. Int J Quantum Chem, 2007, 107: 875-878
|
38 |
Zhou B, Gutman I. Further properties of Zagreb indices. MATCH Commun Math Comput Chem, 2005, 54: 233-239
|
39 |
Zhou B, Trinajstić N. On reciprocal molecular topological index. J Math Chem, 2008, 44: 235-243
|
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