Extremal graphs for the geometric-arithmetic index with given minimum degree

Let G ( k , n ) be the set of connected simple n -vertex graphs with minimum vertex degree k . The geometric-arithmetic index G A ( G ) of a graph G is defined by? G A ( G ) = ? u v 2 d u d v d u + d v , where d ( u ) is the degree of vertex u and the summation extends over all edges u v of G . In this paper we find for k ? ? k 0 ? ,?with k 0 = q 0 ( n - 1 ) , where q 0 ? 0.088 is the unique positive root of the equation q q + q + 3 q - 1 = 0 , extremal graphs in G ( k , n ) for which the geometric-arithmetic index attains its minimum value, or we give a lower bound. We show that when k or n is even, the extremal graphs are regular graphs of degree k .

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