Bounds on the ABC spectral radius of a tree

• Main Authors: Sasmita Barik, Sonu Rani
• Format: Article
• Language: English
• Published: Indonesian Combinatorial Society (InaCombS); Graph Theory and Applications (GTA) Research Centre; University of Newcastle, Australia; Institut Teknologi Bandung (ITB), Indonesia 2020-10-01
• Series: Electronic Journal of Graph Theory and Applications
• Subjects: tree, abc matrix, abc spectral radius, nonnegative matrix
• Online Access: https://www.ejgta.org/index.php/ejgta/article/view/794

<p>Let <em>G</em> be a simple connected graph with vertex set {1,2,...,<em>n</em>} and <em>d_i</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>^th entry is √((<em>d_i+d_j</em>-2)/<em>d_id_i</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>