Sharp Bounds on (Generalized) Distance Energy of Graphs

Given a simple connected graph G, let mml:semantics D(G)mml:semantics be the distance matrix, mml:semantics DL(G)mml:semantics be the distance Laplacian matrix, mml:semanticsDQ(G)mml:semantics be the distance signless Laplacian matrix, and mml:semanticsTr(G)mml:semantics be the vertex transmission diagonal matrix of G. We introduce the generalized distance matrix mml:semanticsD alpha (G)=alpha Tr(G)+(1-alpha )D(G)ml:semantics, where mml:semantics alpha is an element of [0,1]mml:semantics. Noting that mml:semanticsD0(G)=D(G),mml:mspace width="3.33333pt"mml:mspacemml:mspace width="3.33333pt"mml:mspacemml:mspace width="3.33333pt"mml:mspace2mml:mfrac12mml:mfrac>(G)=DQ(G),mml:mspace width="3.33333pt"mml:mspacemml:mspace width="3.33333pt"mml:mspace mml:mspace width="3.33333ptmml:mspace D1(G)=Tr(G)mml:semantics and mml:semanticsD alpha (G)-D beta (G)=(alpha-beta )DL(G)mml:semantics, we reveal that a generalized distance matrix ideally bridges the spectral theories of the three constituent matrices. In this paper, we obtain some sharp upper and lower bounds for the generalized distance energy of a graph G involving different graph invariants. As an application of our results, we will be able to improve some of the recently given bounds in the literature for distance energy and distance signless Laplacian energy of graphs. The extremal graphs of the corresponding bounds are also characterized.