On the smallest eigenvalue of D α-matrix of connected graphs

Abstract

ABSTRACT For a simple connected graph G of order n, let , , and be, respectively, the distance matrix, the diagonal matrix of the vertex transmissions, the distance Laplacian matrix and the distance signless Laplacian matrix of G. For a real number , the generalized distance matrix is defined as . In this paper, we establish relations between the smallest eigenvalues of , and . We obtain sharp upper and lower bounds for the smallest eigenvalue of in terms of various graph parameters like the order n, the diameter d, the Wiener index , the chromatic number , the transmission degrees and the parameter α. We also identify the extremal graphs attaining the given bounds.