Positive semidefiniteness of \(D_{\alpha }\)-matrix on some families of graphs

Let G be a simple connected graph of order n, D(G) the distance matrix of G, and Tr(G) the diagonal matrix of the vertex transmissions of G. The generalized distance matrix of G is defined as \(D_{\alpha }(G)=\alpha Tr(G)+(1-\alpha )D(G)\) $D_{\alpha }(G)=\alpha Tr(G)+(1-\alpha )D(G)$ , \(0\le \alpha \le 1\) $0\le \alpha \le 1$ . It is also commonly referred to as the \(D_\alpha \) $D_{\alpha }$ -matrix of G. Previously, some scholars have studied the positive semidefinite properties of \(D_{\alpha }\) $D_{\alpha }$ -matrix of some special graphs. In this paper, we will study the positive semidefinite properties of \(D_{\alpha }\) $D_{\alpha }$ -matrix of four special graphs, and give the corresponding conditions that \(\alpha \) $\alpha$ needs to satisfy.