On the smallest eigenvalue of Dα-matrix of connected graphs

For a simple connected graph G of order n, let D(G), Tr(G), D-L(G) and D-Q(G) 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 alpha is an element of [0, 1], the generalized distance matrix D-alpha(G) is defined as D-alpha(G) = alpha Tr(G) + (1 - alpha)D(G). In this paper, we establish relations between the smallest eigenvalues of D(G), D-Q(G) and D-alpha(G). We obtain sharp upper and lower bounds for the smallest eigenvalue of D-alpha(G) in terms of various graph parameters like the order n, the diameter d, the Wiener index W(G), the chromatic number chi(G), the transmission degrees and the parameter a. We also identify the extremal graphs attaining the given bounds.