ON THE GENERALIZED DISTANCE EIGENVALUES OF GRAPHS

For a simple connected graph G, the generalized distance matrix D_α(G) is defined as D_α(G) = αT r(G) + (1 − α)D(G), 0 ≤ α ≤ 1. The largest eigenvalue of D_α(G) is called the generalized distance spectral radius or D_α-spectral radius of G. In this paper, we obtain some upper bounds for the generalized distance spectral radius in terms of various graph parameters associated with the structure of graph G, and characterize the extremal graphs attaining these bounds. We determine the graphs with minimal generalized distance spectral radius among the trees with given diameter d and among all unicyclic graphs with given girth. We also obtain the generalized distance spectrum of the square of the cycle and the square of the hypercube of dimension n. We show that the square of the hypercube of dimension n has three distinct generalized distance eigenvalues.