On adjacency-distance spectral radius and spread of graphs

Abstract Let G be a connected graph. The greatest eigenvalue and the spread of the sum of the adjacency matrix and the distance matrix of G are called the adjacency-distance spectral radius and the adjacency-distance spread of G, respectively. Both quantities are used as molecular descriptors in chemoinformatics. We establish some properties for the adjacency-distance spectral radius and the adjacency-distance spread by proposing local grafting operations such that the adjacency-distance spectral radius is decreased or increased. Hence, we characterize those graphs that uniquely minimize and maximize the adjacency-distance spectral radii in several sets of graphs, and determine trees with small adjacency-distance spreads. It transpires that the adjacency-distance spectral radius satisfies the requirements of a branching index.