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 ijth entry is d i +d j −2 d i d j ; if 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.

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“…Estrada [ 33 ] proposed that the atom‐bond connectivity ( ) matrix of graphs in the conjugation with ABC index and defined as Let be the eigenvalues of ABC are called ABC‐eigenvalues with a graph G . He also proposed the ABC spectral radius and energy [ 34 ] and defined as Many researchers computed the ABC spectral radius and energy for different graphs [ 35 , 36 , 37 , 38 , 39 , 40 , 41 ].…”

“…Estrada [33] proposed that the atom‐bond connectivity () matrix of graphs in the conjugation with ABC index and defined as Let be the eigenvalues of ABC are called ABC‐eigenvalues with a graph G . He also proposed the ABC spectral radius and energy [34] and defined as Many researchers computed the ABC spectral radius and energy for different graphs [35–41].…”

“…Many researchers computed the ABC spectral radius and energy for different graphs [35][36][37][38][39][40][41].…”