### Bounds on the ABC spectral radius of a tree

#### Abstract

Let *G* be a simple connected graph with vertex set {1,2,...,*n*} and *d _{i}* denote the degree of vertex

*i*in

*G*. The

*ABC*matrix of

*G*, recently introduced by Estrada, is the square matrix whose

*ij*

^{th}entry is √((

*d*-2)/

_{i}+d_{j}*d*); if

_{i}d_{i}*i*and

*j*are adjacent, and zero; otherwise. The entries in

*ABC*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

*ABC*spectral radius of

*G*in terms of the number of vertices in

*G*. The trees with maximum and minimum

*ABC*spectral radius are characterized. Also, in the class of trees on

*n*vertices, we obtain the trees having first four values of

*ABC*spectral radius and subsequently derive a better upper bound.

#### Keywords

#### Full Text:

PDFDOI: http://dx.doi.org/10.5614/ejgta.2020.8.2.18

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