The spread of generalized reciprocal distance matrix

Abstract

The generalized reciprocal distance matrix $R{D}_{\alpha }(G)$ was defined as $R{D}_{\alpha }(G)=\alpha RT(G)+(1-\alpha )RD(G),0\le \alpha \le 1.$ Let ${\lambda }_1(R{D}_{\alpha }(G))\ge {\lambda }_2(R{D}_{\alpha }(G))\ge \cdots \ge {\lambda }_n(R{D}_{\alpha }(G))$ be the eigenvalues of $R{D}_{\alpha }$ matrix of graphs $G$. Then the $R{D}_{\alpha }$-spread of graph $G$ can be defined as $S_{R{D}_{\alpha }}(G)={\lambda }_1(R{D}_{\alpha }(G))-{\lambda }_n(R{D}_{\alpha }(G))$. In this paper, we first obtain some sharp lower and upper bounds for the $R{D}_{\alpha }$-spread of graphs. Then we determine the lower bounds for the $R{D}_{\alpha }$-spread of bipartite graphs and graphs with given clique number. At last, we give the $R{D}_{\alpha }$-spread of double star graphs. Our results generalize the related results of the reciprocal distance matrix and reciprocal distance signless Laplacian matrix.

14 pages