Distance spectral radius of trees with given matching number

The distance spectral radius ρ(G)ρ(G) of a graph GG is the largest eigenvalue of the distance matrix D(G)D(G). Recently, many researches proposed the use of ρ(G)ρ(G) as a molecular structure descriptor of alkanes. In this paper, we introduce general transformations that decrease distance spectral radius and characterize nn-vertex trees with given matching number mm which minimize the distance spectral radius. The extremal tree A(n,m)A(n,m) is a spur, obtained from the star graph Sn−m+1Sn−m+1 with n−m+1n−m+1 vertices by attaching a pendent edge to each of certain m−1m−1 non-central vertices of Sn−m+1Sn−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≤24n≤24 vertices. In addition, we found the extremal tree that uniquely maximizes the distance spectral radius among nn-vertex trees with perfect matching and fixed maximum degree ΔΔ.