The generalized distance matrix

Abstract Let D ( G ) and D i a g ( T r ) denote the distance matrix and diagonal matrix of the vertex transmissions of a simple connected graph G, respectively. The distance signless Laplacian matrix of G is defined as D Q ( G ) = D i a g ( T r ) + D ( G ) . Heretofore, the spectral properties of D ( G ) and D Q ( G ) have attracted much more attention. In the present paper, we propose to study the convex combinations D α ( G ) of D i a g ( T r ) and D ( G ) , defined as D α ( G ) = α D i a g ( T r ) + ( 1 − α ) D ( G ) , 0 ≤ α ≤ 1 . This study sheds new light on D ( G ) and D Q ( G ) . Some spectral properties of D α ( G ) are given and a few open problems are discussed. Furthermore, we take effort to obtain some upper and lower bounds of spectral radius of D α ( G ) . Finally, the generalized distance spectra of some graphs obtained by operations are also studied.

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