Distance spectral radius of trees with given matching number

Abstract

The distance spectral radius ρ ( G ) of a graph G is the largest eigenvalue of the distance matrix D ( G ) . Recently, many researches proposed the use of ρ ( G ) as a molecular structure descriptor of alkanes. In this paper, we introduce general transformations that decrease distance spectral radius and characterize n -vertex trees with given matching number m which minimize the distance spectral radius. The extremal tree A ( n , m ) is a spur, obtained from the star graph S n − m + 1 with n − m + 1 vertices by attaching a pendent edge to each of certain m − 1 non-central vertices of S n − m + 1 . The resulting trees also minimize the spectral radius of adjacency matrix, Hosoya index, Wiener index and graph energy in the same class of trees. In conclusion, we pose a conjecture for the maximal case based on the computer search among trees on n ≤ 24 vertices. In addition, we found the extremal tree that uniquely maximizes the distance spectral radius among n -vertex trees with perfect matching and fixed maximum degree Δ .