*Article* **Upper and Lower Bounds for the Spectral Radius of Generalized Reciprocal Distance Matrix of a Graph**

**Yuzheng Ma 1, Yubin Gao 2,\* and Yanling Shao <sup>2</sup>**


**\*** Correspondence: ybgao@nuc.edu.cn

**Abstract:** For a connected graph *G* on *n* vertices, recall that the reciprocal distance signless Laplacian matrix of *G* is defined to be *RQ*(*G*) = *RT*(*G*) + *RD*(*G*), where *RD*(*G*) is the reciprocal distance matrix, *RT*(*G*) = *diag*(*RT*1, *RT*2, ... , *RTn*) and *RTi* is the reciprocal distance degree of vertex *vi*. In 2022, generalized reciprocal distance matrix, which is defined by *RDα*(*G*) = *αRT*(*G*)+(1 − *α*)*RD*(*G*), *α* ∈ [0, 1], was introduced. In this paper, we give some bounds on the spectral radius of *RDα*(*G*) and characterize its extremal graph. In addition, we also give the generalized reciprocal distance spectral radius of line graph *L*(*G*).

**Keywords:** graph; generalized reciprocal distance matrix; reciprocal distance signless Laplacian matrix; spectral radius

**MSC:** 05C50; 05C12; 15A18
