Distance spectral radius of trees with given matching number

The distance spectral radius rho(G) of a graph G is the largest eigenvalue of the distance matrix D(G). Recently, many researches proposed the use of rho(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 extrema! tree A(n, m) is a spur, obtained from the star graph Sn-m+1, with n - m + 1 vertices by attaching a pendent edge to each of certain m - 1 non-central vertices of Sn-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 Delta. (C) 2010 Elsevier B.V. All rights reserved.