On the spread of the geometric-arithmetic matrix of graphs

Abstract

In a graph G, if di is the degree of a vertex vi, the geometric-arithmetic matrix GA(G) is a square matrix whose [Formula: see text]-th entry is [Formula: see text] whenever vertices i and j are adjacent and 0 otherwise. The set of all eigenvalues of GA(G) including multiplicities is known as the geometric-arithmetic spectrum of G. The difference between the largest and the smallest geometric-arithmetic eigenvalue is called the geometric-arithmetic spread [Formula: see text] of G. In this article, we investigate some properties of [Formula: see text] We obtain lower and upper bounds of [Formula: see text] and show the existence of graphs for which equality holds. Further, [Formula: see text] is computed for various graph operations.

Keywords

• Adjacency matrix
• spectral radius
• geometric-arithmetic index
• geometric-arithmetic spectrum
• spread
• 05C50