New bounds of extended energy of a graph

The extended adjacency matrix of a graph with $n$ vertices is a real symmetric matrix of order $n\times n$ whose $(i,j)$-th entry is the average of the ratio of the degree of the vertex $i$ to that of the vertex $j$ and its reciprocal when $i,j$ are adjacent and zero otherwise. The aggregate of absolute eigenvalues of the extended adjacency matrix is termed the extended energy. In this paper, the concept of extended vertex energy is introduced, and some bounds of extended vertex energy are obtained. From there, we establish some new upper bounds of the extended energy of a graph involving order, size, largest, and smallest degree. We show that those are improvements of some existing bounds. Through direct manipulation, we have also established some more upper and lower bounds of extended energy, which are either better or incomparable with the existing bounds. Finally, some improved bounds of Nordhaus-Gaddum-type are found.