| Summary: | <p>Let <em>G</em> be a simple connected graph with vertex set {1,2,...,<em>n</em>} and <em>d<sub>i</sub></em> denote the degree of vertex <em>i</em> in <em>G</em>. The <em>ABC</em> matrix of <em>G</em>, recently introduced by Estrada, is the square matrix whose <em>ij</em><sup>th</sup> entry is √((<em>d<sub>i</sub>+d<sub>j</sub></em>-2)/<em>d<sub>i</sub>d<sub>i</sub></em>); if <em>i</em> and <em>j</em> are adjacent, and zero; otherwise. The entries in <em>ABC</em> matrix represent the probability of visiting a nearest neighbor edge from one side or the other of a given edge in a graph. In this article, we provide bounds on <em>ABC</em> spectral radius of <em>G</em> in terms of the number of vertices in <em>G</em>. The trees with maximum and minimum <em>ABC</em> spectral radius are characterized. Also, in the class of trees on <em>n</em> vertices, we obtain the trees having first four values of <em>ABC</em> spectral radius and subsequently derive a better upper bound.</p>
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