On the Generalized Distance Energy of Graphs

The generalized distance matrix D alpha(G) of a connected graph G is defined as D alpha(G)=alpha Tr(G)+(1-alpha)D(G), where 0 <=alpha <= 1, D(G) is the distance matrix and Tr(G) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy ED alpha(G). Some new upper and lower bounds for the generalized distance energy ED alpha(G) of G are established based on parameters including the Wiener index W(G) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.