Some upper and lower bounds for $D_{\alpha}$-energy of graphs

Abstract

The generalized distance matrix of a connected graph $G$, denoted by $D_{\alpha}(G)$, is defined as $D_{\alpha}(G)=\alpha Tr(G)+(1-\alpha)D(G), ~~~~ 0\leq \alpha\leq 1$. Here, $D(G)$ is the distance matrix and $Tr(G)$ represents the vertex transmissions. Let $\partial_{1}\geq \partial_{2}\geq \cdots \geq \partial_{n}$ be the eigenvalues of $D_{\alpha}(G)$ and let $W(G)$ be the Wiener index. The generalizeddistance energy of $G$ can be defined as $E^{D_{\alpha}}(G)=\displaystyle\sum_{i=1}^{n}\left|\partial_i-\frac{2\alpha W(G)}{n}\right|$. In this paper, we develop some new theory regarding the generalized distance energy $E^{D_{\alpha}}(G)$ for a connected graph $G$. We obtain some sharp upper and lower bounds for$E^{D_{\alpha}}(G)$ connecting a wide range of parameters in graph theory including the maximum degree $\Delta$, the Wiener index $W(G)$, the diameter $d$, the transmission degrees, and the generalized distance spectral spread $D_{\alpha}S(G)$. We characterized the special graph classes that attain the bounds.