On the atom-bond sum-connectivity spectral radius of trees

The atom-bond sum-connectivity (ABS) matrix of a graph G with n vertices is the square matrix of order n whose (i, j)entry is equal to (di + dj -2)/(di + dj) if the i-th vertex and the j-th vertex of G are adjacent, and 0 otherwise, where di is the degree of the i-th vertex of G.The ABS spectral radius of G is the largest eigenvalue of the ABS matrix of G, which is denoted by λABS(G).In this paper, the chemical importance of the ABS spectral radius is investigated and it is found that this new spectral parameter is useful in predicting certain physicochemical properties of molecules with an accuracy higher than the atom-bond sum-connectivity index.Also, for any tree Tn with n ≥ 5 vertices, it is proved that λABS(Pn) ≤ λABS(Tn) ≤ λABS(Sn), with equality in the left (respectively, right) inequality if and only if Tn is isomorphic to the path Pn (respectively, the star Sn).