### G-2016-39

# Comparing the geometric-arithmetic index and the spectral radius of graphs

## Mustapha Aouchiche and Pierre Hansen

The geometric-arithmetic index `\(GA\)`

of a graph `\(G\)`

is the sum of ratios, over all edges of `\(G\)`

, of the geometric mean to the arithmetic mean of the end vertices degrees of an edge. The spectral radius `\(\lambda_1\)`

of `\(G\)`

is the largest eigenvalue of its adjacency matrix. These two parameters are known to be used as molecular descriptors in chemical graph theory.

In the present paper, we compare `\(GA\)`

and `\(\lambda_1\)`

of a connected graph with given order. We prove, among other results, upper and lower bounds on the ratio `\(GA/\lambda_1\)`

as well as a lower bound on the ratio `\(GA/\lambda_1^2\)`

. In addition, we characterize all extremal graphs corresponding to each of these bounds.

Published **June 2016**
,
10 pages