The spread of generalized reciprocal distance matrix

Abstract

The generalized reciprocal distance matrix $RD_{\alpha}(G)$ was defined as $RD_{\alpha}(G)=\alpha RT(G)+(1-\alpha)RD(G),\quad 0\leq \alpha \leq 1.$ Let $\lambda_{1}(RD_{\alpha}(G))\geq \lambda_{2}(RD_{\alpha}(G))\geq \cdots \geq \lambda_{n}(RD_{\alpha}(G))$ be the eigenvalues of $RD_{\alpha}$ matrix of graphs $G$. Then the $RD_{\alpha}$-spread of graph $G$ can be defined as $S_{RD_{\alpha}}(G)=\lambda_{1}(RD_{\alpha}(G))-\lambda_{n}(RD_{\alpha}(G))$. In this paper, we first obtain some sharp lower and upper bounds for the $RD_{\alpha}$-spread of graphs. Then we determine the lower bounds for the $RD_{\alpha}$-spread of bipartite graphs and graphs with given clique number. At last, we give the $RD_{\alpha}$-spread of double star graphs. Our results generalize the related results of the reciprocal distance matrix and reciprocal distance signless Laplacian matrix.