The spread of generalized reciprocal distance matrix

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

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