On the first geometric-arithmetic index of graphs

Let G be a simple connected graph and d"i be the degree of its ith vertex. In a recent paper [D. Vukicevic, B. Furtula, Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges, J. Math. Chem. 46 (2009) 1369-1376] the ''first geometric-arithmetic index'' of a graph G was defined as GA"1=@?d"id"j(d"i+d"j)/2 with summation going over all pairs of adjacent vertices. We obtain lower and upper bounds on GA"1 and characterize graphs for which these bounds are best possible. Moreover, we discuss the effect on GA"1 of inserting an edge into a graph.

[1]  Heping Zhang,et al.  Kirchhoff index of composite graphs , 2009, Discret. Appl. Math..

[2]  Peter Dankelmann,et al.  On the degree distance of a graph , 2009, Discret. Appl. Math..

[3]  Kinkar Chandra Das,et al.  Atom-bond connectivity index of graphs , 2010, Discret. Appl. Math..

[4]  Ardeshir Dolati,et al.  Szeged index, edge Szeged index, and semi-star trees , 2010, Discret. Appl. Math..

[5]  Tomaz Pisanski,et al.  Use of the Szeged index and the revised Szeged index for measuring network bipartivity , 2010, Discret. Appl. Math..

[6]  D. Vukicevic,et al.  Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges , 2009 .

[7]  I. Gutman,et al.  Survey on Geometric-Arithmetic Indices of Graphs , 2011 .

[8]  G. Pólya,et al.  Problems and Theorems in Analysis I: Series. Integral Calculus. Theory of Functions , 1976 .

[9]  Xiaochun Cai,et al.  Reciprocal complementary Wiener numbers of trees, unicyclic graphs and bicyclic graphs , 2009, Discret. Appl. Math..

[10]  Ting Chen,et al.  Comparing the Zagreb indices for graphs with small difference between the maximum and minimum degrees , 2009, Discret. Appl. Math..

[11]  Sang-Gu Lee,et al.  Comparing Zagreb indices for connected graphs , 2010, Discret. Appl. Math..

[12]  Roberto Todeschini,et al.  Molecular descriptors for chemoinformatics , 2009 .

[13]  G. Pólya,et al.  Problems and theorems in analysis , 1983 .

[14]  Kinkar Ch. Das,et al.  Estimating the Wiener Index by Means of Number of Vertices, Number of Edges, and Diameter , 2010 .

[15]  Ante Graovac,et al.  Atom-bond connectivity index of trees , 2009, Discret. Appl. Math..