Group for Research in Decision Analysis

G-2016-39

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

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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