On the smallest eigenvalue of Dα-matrix of connected graphs. Issue 21 (20th December 2022) - Alhevaz, Abdollah; Baghipur, Maryam; Pirzada, Shariefuddin
ABSTRACT: 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 α ∈ [0, 1], the generalized distance matrix D α ( G ) is defined as D α ( G ) = αTr ( G ) + (1 − α ) D ( G ). In this paper, we establish relations between the smallest eigenvalues of D ( G ), D Q ( G ) and D α ( G ). We obtain sharp upper and lower bounds for the smallest eigenvalue of D α ( G ) in terms of various graph parameters like the order n, the diameter d, the Wiener index W ( G ), the chromatic number χ ( G ), the transmission degrees and the parameter α . We also identify the extremal graphs attaining the given bounds.