Sharp upper bounds for Zagreb indices of bipartite graphs with a given diameter

Abstract For a (molecular) graph, the first Zagreb index M 1 is equal to the sum of squares of the vertex degrees, and the second Zagreb index M 2 is equal to the sum of products of the degrees of a pair of adjacent vertices. In this work, we study the Zagreb indices of bipartite graphs of order n with diameter d and sharp upper bounds are obtained for M 1 ( G ) and M 2 ( G ) with G ∈ ℬ ( n , d ) , where ℬ ( n , d ) is the set of all the n -vertex bipartite graphs with diameter d . Furthermore, we study the relationship between the maximal Zagreb indices of graphs in ℬ ( n , d ) and the diameter d . As a consequence, bipartite graphs with the largest, second-largest and smallest Zagreb indices are characterized.

[1]  D. Cvetkovic,et al.  Graph theory and molecular orbitals , 1974 .

[2]  A. Balaban,et al.  Topological Indices for Structure-Activity Correlations , 1983, Steric Effects in Drug Design.

[3]  I. Gutman,et al.  Graph theory and molecular orbitals. XII. Acyclic polyenes , 1975 .

[4]  Roberto Todeschini,et al.  Handbook of Molecular Descriptors , 2002 .

[5]  Nenad Trinajstić,et al.  Trees with maximal second Zagreb index and prescribed number of vertices of the given degree , 2008 .

[6]  Rossella Petreschi,et al.  (n,e)-Graphs with maximum sum of squares of degrees* , 1999 .

[7]  Kinkar Chandra Das,et al.  Maximizing the sum of the squares of the degrees of a graph , 2004, Discret. Math..

[8]  Shuchao Li,et al.  On the Maximum Zagreb Indices of Graphs with k Cut Vertices , 2010 .

[9]  Shuchao Li,et al.  Sharp bounds for Zagreb indices of maximal outerplanar graphs , 2011, J. Comb. Optim..

[10]  N. Trinajstic,et al.  The Zagreb Indices 30 Years After , 2003 .

[11]  Lingli Sun,et al.  The second Zagreb index of acyclic conjugated molecules , 2008 .

[12]  Shenggui Zhang,et al.  Extreme values of the sum of squares of degrees of bipartite graphs , 2009, Discret. Math..

[13]  Lihua Feng,et al.  Zagreb, Harary and hyper-Wiener indices of graphs with a given matching number , 2010, Appl. Math. Lett..

[14]  Shuchao Li,et al.  On the maximum and minimum Zagreb indices of graphs with connectivity at most k , 2010, Appl. Math. Lett..

[15]  Sebastian M. Cioab Note: Sums of powers of the degrees of a graph , 2006 .

[16]  Ying Wang,et al.  BIPARTITE GRAPHS WITH EXTREME VALUES OF THE FIRST GENERAL ZAGREB INDEX , 2010 .

[17]  Ivan Gutman,et al.  Two early branching indices and the relation between them , 2002 .

[18]  I. Gutman,et al.  Graph theory and molecular orbitals. Total φ-electron energy of alternant hydrocarbons , 1972 .

[19]  Mei Lu,et al.  On the connectivity index of trees , 2008 .

[20]  H. C. Longuet-Higgins,et al.  Some Studies in Molecular Orbital Theory I. Resonance Structures and Molecular Orbitals in Unsaturated Hydrocarbons , 1950 .

[21]  Dan Stefanica,et al.  Minimizer graphs for a class of extremal problems , 2002 .

[22]  Huiqing Liu,et al.  Trees of extremal connectivity index , 2006, Discret. Appl. Math..

[23]  Huiqing Liu,et al.  Sharp Bounds for the Second Zagreb Index of Unicyclic Graphs , 2007 .

[24]  Ruifang Liu,et al.  On the spectral radius of bipartite graphs with given diameter , 2009 .

[25]  Xia Hu,et al.  On the Extremal Zagreb Indices of Graphs with Cut Edges , 2010 .

[26]  Dominique de Caen,et al.  An upper bound on the sum of squares of degrees in a graph , 1998, Discret. Math..