Group for Research in Decision Analysis

# 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.

, 10 pages