**4. Bounds of** *ρ***(***RDα***(***G***)) of Line Graph** *L***(***G***)**

The line graph *L*(*G*) of *G* is the graph whose vertices correspond to the edges of *G*, and two vertices of *L*(*G*) are adjacent if and only if the corresponding edges of *G* are adjacent. In this section, we give the bounds of the spectral radius of the generalized reciprocal distance matrix of *L*(*G*).

**Theorem 9.** *Let graph G have n vertices and m edges, and the degree of vertex vi be recorded as di. If diam*(*G*) ≤ 2 *and graphs Fi, i* = 1, 2, 3 *in Lemma 10 are not induced subgraphs of G, then*

$$\rho(RD\_a(L(G))) \ge \frac{\frac{1}{2}(m^2 - 3m + \sum\_{i=1}^n d\_i^2)}{m}.$$

**Proof.** If *diam*(*G*) ≤ 2, the *i*-th row element of *RDα*(*G*) is composed of { <sup>1</sup> <sup>2</sup> *α*(*n* + *di* − 1),(1 − *α*)*di* , <sup>1</sup> <sup>2</sup> (<sup>1</sup> <sup>−</sup> *<sup>α</sup>*)[*n*−*di*−1] }, which can be obtained from Lemma 9

$$\rho\left(R D\_{\mathbf{z}}\left(L(G)\right)\right) \ge \frac{\mathbf{e}^T R D\_{\mathbf{z}}(G) \mathbf{e}}{\mathbf{e}^T \mathbf{e}} = \frac{\sum\_{i=1}^n \frac{1}{2} (n + d\_i - 1)}{n} = \frac{\frac{1}{2} (n^2 + 2m - n)}{n}.$$

Hence, line graph *L*(*G*) has *n*<sup>1</sup> = *m* vertices and *m*<sup>1</sup> = <sup>1</sup> 2 *n* ∑ *i*=1 *d*2 *<sup>i</sup>* − *m* edges. Because graphs *Fi*, *i* = 1, 2, 3 are not induced subgraphs of *G*, from Lemma 10, *diam*(*L*(*G*)) ≤ 2, then

$$\begin{aligned} \rho(RD\_\pi(L(G))) &\geq \frac{\frac{1}{2}(n\_1^2 + 2m\_1 - n\_1)}{n\_1} \\ &= \frac{\frac{1}{2}[m^2 + 2(\frac{1}{2}\sum\_{i=1}^n d\_i^2 - m) - m]}{m} \\ &= \frac{\frac{1}{2}(m^2 - 3m + \sum\_{i=1}^n d\_i^2)}{m} .\end{aligned}$$

**Theorem 10.** *Let graph G be r-regular graph with n vertices, and graphs Fi, i* = 1, 2, 3 *be not-induced subgraphs of G. Then*

$$
\rho(RD\_\mathfrak{a}(L(G))) \ge \frac{nr}{4} + r - 3.
$$

**Proof.** Let graph *G* be *r*-regular graph with *n* vertices, the number of edges in graph *G* is *m* = *nr* <sup>2</sup> , *di* = *deg*(*vi*) = *r*. It is proved by Theorem 9.

**Theorem 11.** *Let the vertices set and edges set of G be V*(*G*) = {*v*1, *v*2, ... , *vn*} *and E*(*G*) = {*e*1,*e*2,...,*em*}*, deg*(*ei*) *represent the number of edges adjacent to edge ei. Then,*

$$\rho\left(RD\_a(L(G)) \le \max\_{1 \le i \le m} \left\{ \frac{1}{2} (m - \deg(e\_i) - 1) \right\} \right)$$

**Proof.** Let *e* = *uv* be an edge of *G*. Then, the degree of vertex *e* ∈ *V*(*L*(*G*)) is *degL*(*G*)(*e*) = *degG*(*u*) + *degG*(*v*) − 2.

In graph *G*, if edge *e* = *uv* is adjacent to *deg*(*u*) + *deg*(*v*) − 2 = *deg*(*e*), then denoted |*Ee*| = *m* − 1 − *deg*(*e*) as the number of edges which are not adjacent to edge *e*. Therefore, in the graph *L*(*G*), there are |*Ee*| vertices, and their distance from vertex *e* is greater than 1. Thus, the maximum element of generalized reciprocal distances matrix of the corresponding vertices should be <sup>1</sup> <sup>2</sup> (1 − *α*). We can get

$$\begin{aligned} S\_i(RD\_a(L(G))) &\le \frac{1}{2}(1-a)(m-1-\deg(e\_i)) \\ &+ (1-a)\deg(e\_i) + a(\frac{1}{2}m - \frac{1}{2} + \frac{1}{2}\deg(e\_i)) \\ &= \frac{1}{2}(m-\deg(e\_i)-1). \end{aligned}$$

By Lemma 4, *ρ*(*RDα*(*L*(*G*))) ≤ max 1≤*i*≤*m* { 1 <sup>2</sup> (*m* − *deg*(*ei*) − 1)}.

#### **5. Conclusions**

In this paper, we find some bounds for the spectral radius of the generalized reciprocal distance matrix of a simple undirected connected graph *G*, and we also give the generalized reciprocal distance spectral radius of line graph *L*(*G*). The graphs for which those bounds are attained are characterized.

**Author Contributions:** Investigation, Y.M., Y.G. and Y.S.; writing—original draft preparation, Y.M.; writing—review and editing, Y.M., Y.G.; All authors have read and agreed to the published version of the manuscript.

**Funding:** Research was supported by Shanxi Scholarship Council of China (No. 201901D211227).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** All data generated or analyzed during this study are included in this published article.

**Acknowledgments:** The authors are grateful to the anonymous referees for helpful suggestions and valuable comments, which led to an improvement of the original manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

