On the geometric-arithmetic index of a graph

Abstract Very recently, Aouchiche and Hansen gave an upper bound on the ratio of G A ∕ χ of the geometric–arithmetic index G A ( G ) and the chromatic number χ ( G ) of a graph. Furthermore, they proposed several conjectures on the relation between geometric–arithmetic index and chromatic number, and clique number. In this note, we disprove four of those conjectures. In addition, we present a sufficient condition for an edge e ∈ E ( G ) with G A ( G − e ) G A ( G ) .

[1]  Xiaojing Yang,et al.  Randić index and coloring number of a graph , 2014, Discret. Appl. Math..

[2]  Kinkar Chandra Das,et al.  On the first geometric-arithmetic index of graphs , 2011, Discret. Appl. Math..

[3]  Béla Bollobás,et al.  Graphs of Extremal Weights , 1998, Ars Comb..

[4]  M. Randic.  Characterization of molecular branching , 1975 .

[5]  Manouchehr Zaker,et al.  More bounds for the Grundy number of graphs , 2015, J. Comb. Optim..

[6]  M. Rodr,et al.  On the Geometric{Arithmetic Index , 2015 .

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

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

[9]  Ljiljana Pavlovic,et al.  Extremal graphs for the geometric-arithmetic index with given minimum degree , 2014, Discret. Appl. Math..

[10]  Jose Maria Sigarreta,et al.  Spectral properties of geometric-arithmetic index , 2016, Appl. Math. Comput..

[11]  Hanyuan Deng,et al.  On the harmonic index and the chromatic number of a graph , 2013, Discret. Appl. Math..

[12]  Bo Zhou,et al.  On geometric-arithmetic index , 2010 .

[13]  Pierre Hansen,et al.  Variable neighborhood search for extremal graphs. 23. On the Randic index and the chromatic number , 2009, Discret. Math..

[14]  Mustapha Aouchiche,et al.  The geometric-arithmetic index and the chromatic number of connected graphs , 2016, Discret. Appl. Math..

[15]  Qing Cui,et al.  On a relation between the atom-bond connectivity and the first geometric-arithmetic indices , 2015, Discret. Appl. Math..

[16]  G. Fath-Tabar,et al.  Some Bounds on GA1 Index of Graphs , 2011 .

[17]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .