Merging the Spectral Theories of Distance Estrada and Distance Signless Laplacian Estrada Indices of Graphs

Suppose that G is a simple undirected connected graph. Denote by D(G) the distance matrix of G and by Tr(G) the diagonal matrix of the vertex transmissions in G, and let α∈[0,1] . The generalized distance matrix Dα(G) is defined as Dα(G)=αTr(G)+(1−α)D(G) , where 0≤α≤1 . If ∂1≥∂2≥…≥∂n are the eigenvalues of Dα(G) ; we define the generalized distance Estrada index of the graph G as DαE(G)=∑ni=1e(∂i−2αW(G)n), where W(G) denotes for the Wiener index of G. It is clear from the definition that D0E(G)=DEE(G) and 2D12E(G)=DQEE(G) , where DEE(G) denotes the distance Estrada index of G and DQEE(G) denotes the distance signless Laplacian Estrada index of G. This shows that the concept of generalized distance Estrada index of a graph G merges the theories of distance Estrada index and the distance signless Laplacian Estrada index. In this paper, we obtain some lower and upper bounds for the generalized distance Estrada index, in terms of various graph parameters associated with the structure of the graph G, and characterize the extremal graphs attaining these bounds. We also highlight relationship between the generalized distance Estrada index and the other graph-spectrum-based invariants, including generalized distance energy. Moreover, we have worked out some expressions for DαE(G) of some special classes of graphs.