**1. Introduction**

In this paper, all graphs considered are finite, simple, and connected. Let *G* be such a graph with vertex set *V*(*G*) = {*v*1, *v*2, ... , *vn*} and edge set *E*(*G*), where |*V*(*G*)| = *n* and |*E*(*G*)| = *m*. Let *dvi* denote the degree of vertex *vi*, which is simply written as *di*. *N*(*vi*) denote the neighbor set of *vi*. The distance between vertices *vi* and *vj* in *G* is the length of the shortest path connecting *vi* to *vj*, which is denoted as *d*(*vi*, *vj*). We use the notation *dij* instead of *d*(*vi*, *vj*). The diameter of *G*, denoted by *diam*(*G*), is the maximum distance between any pair of vertices of *G*. The Harary matrix of *G*, which is also called the reciprocal distance matrix, is an *n* × *n* matrix defined as [1]

$$RD\_{i,j} = \begin{cases} \frac{1}{d(v\_{i'}v\_j)}, & \text{if } i \neq j, \\ 0, & \text{if } i = j. \end{cases}$$

Henceforth, we consider *i* = *j* for *d*(*vi*, *vj*).

The transmission of vertex *vi*, denoted by *TrG*(*vi*) or *Tri*, is defined to be the sum of the distances from *vi* to all vertices in *G* , that is, *TrG*(*vi*) = *Tri* = ∑ *u*∈*V*(*G*) *d*(*u*, *vi*). A graph *G* is said to be *k*-transmission regular graph if *TrG*(*v*) = *k* for each *v* ∈ *V*(*G*). Transmission

**Definition 1.** *Let G be a graph with V*(*G*) = {*v*1, *v*2, ... , *vn*}*. The reciprocal distance degree of a vertex v, denoted by RTrG*(*v*)*, is given by*

of a vertex *v* is also called the distance degree or the first distance degree of *v*.

$$R\!Tr\_G(v) = \sum\_{\substack{\mu \in V(G), \mu \neq v}} \frac{1}{d(\mu, v)}.$$

**Citation:** Ma, Y.; Gao, Y.; Shao, Y. Upper and Lower Bounds for the Spectral Radius of Generalized Reciprocal Distance Matrix of a Graph. *Mathematics* **2022**, *10*, 2683. https://doi.org/10.3390/ math10152683

Academic Editors: Irina Cristea and Hashem Bordbar

Received: 28 June 2022 Accepted: 27 July 2022 Published: 29 July 2022

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*Let RT*(*G*) *be the n* × *n diagonal matrix defined by RTi*,*<sup>i</sup>* = *RTrG*(*vi*)*.*

Sometimes we use the notation *RTi* instead of *RTrG*(*vi*) for *i* = 1, . . . , *n*.

**Definition 2.** *A graph G is called a k-reciprocal distance degree regular graph if RTi* = *k for all i* ∈ {1, 2, . . . , *n*}.

The Harary index of a graph *G*, denoted by *H*(*G*), is defined in [1] as

$$H(G) = \frac{1}{2} \sum\_{i=1}^{n} \sum\_{j=1}^{n} RD\_{i,j} = \frac{1}{2} \sum\_{\substack{\mu, \nu \in V(G), \mu \neq \nu}} \frac{1}{d(\mu, \nu)}.$$

Clearly,

$$H(G) = \frac{1}{2} \sum\_{i=1}^{n} RT\_i.$$

In [2], Bapat and Panda defined the reciprocal distance Laplacian matrix as *RL*(*G*) = *RT*(*G*) − *RD*(*G*). It was proved that, given a connected graph *G* of order *n*, the spectral radius of its reciprocal distance Laplacian matrix *ρ*(*RL*(*G*)) ≤ *n* if and only if its complement graph, denoted by *G*, is disconnected. In [3], Alhevaz et al. defined the reciprocal distance signless Laplacian matrix as *RQ*(*G*) = *RT*(*G*) + *RD*(*G*). Recently, the lower and upper bounds of the spectral radius of the reciprocal distance matrices and reciprocal distance signless Laplacian matrices of graphs were given in [3–6], respectively.

In [7], the author, using the convex linear combinations of the matrices *RT*(*G*) and *RD*(*G*), introduces a new matrix, that is generalized reciprocal distance matrix, denoted by *RDα*(*G*), which is defined by

$$RD\_a(G) = \alpha RT(G) + (1 - \alpha)RD(G), \; 0 \le \alpha \le 1.$$

Since *RD*0(*G*) = *RD*(*G*), *RD*<sup>1</sup> 2 (*G*) = <sup>1</sup> <sup>2</sup>*RQ*(*G*) and *RD*1(*G*) = *RT*(*G*), then *RD*<sup>1</sup> 2 (*G*) and *RQ*(*G*) have the same spectral properties. To this extent these matrices *RD*(*G*), *RT*(*G*), and *RQ*(*G*) may be understood from a completely new perspective, and some interesting topics arise. For the these matrices *RD*(*G*), *RT*(*G*), and *RQ*(*G*), some spectral extremal graphs with fixed structure parameters have been characterized in [8,9]. It is natural to ask whether these results can be generalized to *RDα*(*G*).

Since *RDα*(*G*) is real symmetric matrics, we can denoted *λ*1(*RDα*(*G*)) ≥ *λ*2(*RDα*(*G*)) ≥ ··· ≥ *λn*(*RDα*(*G*)) to the eigenvalues of *RDα*(*G*). The maximum eigenvalue *λ*1(*RDα*(*G*)) is called the spectral radius of the matrix *RDα*(*G*), denoted by *ρ*(*RDα*(*G*)).

This paper is organized as follows. In Section 2, we give some definitions, notations, and lemmas of generalized reciprocal distance matrix. In Section 3, we give the upper and lower bounds of the spectral radius of the generalized reciprocal distance matrix *RDα*(*G*) by using the reciprocal distance degree and the second reciprocal distance degree. In Section 4, we give the bounds of the spectral radius of the generalized reciprocal distance matrix of *L*(*G*), where *L*(*G*) is the line graph of graph *G*.

#### **2. Lemmas**

In this section, we give some definitions, notations, and lemmas to prepare for subsequent proofs.

**Definition 3.** *Let G be a graph with V*(*G*) = {*v*1, *v*2, ... , *vn*}*, the reciprocal distance matrix RD*(*G*) *and the reciprocal distance degree sequence* {*RT*1, *RT*2, ... , *RTn*}. *Then the second reciprocal distance degree of a vertex vi, denoted by Ti, is given by*

$$T\_i = \sum\_{j=1, j \neq i}^{n} \frac{1}{d\_{i,j}} RT\_j... $$

**Definition 4.** *A graph G is called a pseudo k-reciprocal distance degree regular graph if Ti RTi* = *<sup>k</sup> for all i* ∈ {1, 2, . . . , *n*}.

**Definition 5.** *The Frobenius norm of an n* × *n matrix M* = (*mi*,*j*) *is*

$$\|M\|\_F = \sqrt{\sum\_{i=1}^n \sum\_{j=1}^n |m\_{i,j}|^2}.$$

*We recall that, if M is a normal matrix then M* <sup>2</sup> *<sup>F</sup>* <sup>=</sup> *<sup>n</sup>* ∑ *i*=1 |*λi*(*M*)| <sup>2</sup> *where λ*1(*M*), ... , *λn*(*M*) *are the eigenvalues of M. In particular, RDα*(*G*) <sup>2</sup> *<sup>F</sup>* <sup>=</sup> *<sup>n</sup>* ∑ *i*=1 | *λi*(*RDα*(*G*)) | 2 .

**Lemma 1** ([6])**.** *Let G be a graph of order n with reciprocal distance degree sequence* {*RT*1, *RT*2,..., *RTn*} *and second reciprocal distance degree sequence* {*T*1, *T*2,..., *Tn*}*. Then*

$$T\_1 + T\_2 + \dots + T\_n = RT\_1^2 + RT\_2^2 + \dots + RT\_n^2.$$

**Lemma 2** (Perron–Frobenius theorem [10])**.** *If A is a non-negative matrix of order n, then its spectral radius ρ*(*A*) *is an eigenvalue of A and it has an associated non-negative eigenvector. Furthermore, if A is irreducible, then ρ*(*G*) *is a simple eigenvalue of A with an associated positive eigenvector.*

**Lemma 3** ([7])**.** *Let G be a graph with n* ≥ 2 *vertices and Harary index H*(*G*)*. Then*

$$
\rho(RD\_{\mathfrak{A}}(G)) \ge \frac{2H(G)}{n}.
$$

*The equality holds if and only if G is a reciprocal distance degree regular graph.*

**Lemma 4** ([11])**.** *Let A* = (*ai*,*j*) *be an n* × *n nonnegative matrix with spectral radius ρ*(*A*) *and row sums S*1(*A*), *S*2(*A*),..., *Sn*(*A*). *Then,*

$$\min\_{1 \le i \le n} S\_i(A) \le \rho(A) \le \max\_{1 \le i \le n} S\_i(A).$$

*Moreover, if A is an irreducible matrix, then equality holds on either side (and hence both sides) of the equality if and only if all row sums of A are all equal.*

**Lemma 5** ([6])**.** *Let G be a graph on n vertices. Let RTmax and RTmin be the maximum and the minimum reciprocal distance degree of G, respectively. Then, for any vi* ∈ *V*(*G*)*,*

$$2H(G) + (RT\_{\text{max}} - 1)RT\_i - (n - 1)RT\_{\text{max}} \le T\_i \le 2H(G) + (RT\_{\text{min}} - 1)RT\_i - (n - 1)RT\_{\text{min}}.$$

**Lemma 6** (Cauchy alternating theorem [12])**.** *Let A be a real symmetric matrix of order n and B be a principal submatrix of order m of A. Suppose A has eigenvalues λ*<sup>1</sup> ≥ *λ*<sup>2</sup> ≥···≥ *λn, and B has eigenvalues β*<sup>1</sup> ≥ *β*<sup>2</sup> ≥···≥ *βm. Then, for all k* = 1, 2, . . . , *m, λn*−*m*+*<sup>k</sup>* ≤ *β<sup>k</sup>* ≤ *λk.*

**Lemma 7.** *Let G be a graph on n* ≥ 2 *vertices with* 0 ≤ *α* < 1*. The G has exactly two distinct generalized reciprocal distance eigenvalues if and only if G is a complete graph. In particular, ρ*(*RDα*(*Kn*)) = *n* − 1 *and λi*(*RDα*(*Kn*)) = *αn* − 1 *for i* = 2, 3, . . . , *n.*

**Proof.** Let *n* ≥ 2. Clearly, the spectrum of the generalized reciprocal distance matrix of the complete graph *Kn* is {*n* − 1,(*αn* − 1)[*n*−1] }.

Let *G* be a graph with generalized reciprocal distance matrix *RDα*(*G*). If *G* has exactly two distinct *RDα*-eigenvalues, then *λ*1(*RDα*(*G*)) > *λ*2(*RDα*(*G*)). Since *G* is a connected graph and *RDα*(*G*) is an irreducible matrix. Then, from Lemma 2, *λ*1(*RDα*(*G*)) = *ρ*(*RDα*(*G*)) is the greatest and simple eigenvalue of *RDα*(*G*). Thus, the algebraic multiplicity of *λ*2(*RDα*(*G*)) is *n* − 1, i.e.,

$$
\lambda\_2(R D\_\mathfrak{a}(G)) = \lambda\_3(R D\_\mathfrak{a}(G)) = \dots = \lambda\_n(R D\_\mathfrak{a}(G)).\tag{1}
$$

Now, to prove that *G* = *Kn*, we show that the diameter of *G* is 1. That is, we prove that *G* does not contain an shortest path *Pk*, for *k* ≥ 3.

We suppose that *G* contains an induced shortest path *Pk*, *k* ≥ 3. Let *B* be the principal submatrix of *RDα*(*G*) indexed by the vertices in *Pk*. Then by Lemma 6, we have

$$
\lambda\_i(R D\_\mathfrak{a}(G)) \ge \lambda\_i(B) \ge \lambda\_{i+n-k}(R D\_\mathfrak{a}(G)), i = 1, 2, \dots, k.
$$

Using the equalities given in (1), we obtain *λ*2(*RDα*(*G*)) ≥ *λ*2(*B*) ≥ *λ*3(*B*) ≥ ··· ≥ *λk*(*B*) ≥ *λp*(*RDα*(*G*)) = *λ*2(*RDα*(*G*)). Thus, for *k* ≥ 3, the matrix *B* = (*RDα*(*Pk*)) has at most two different eigenvalues. By definition, we can get the generalized reciprocal distance matrix of *P*3, that is

$$RD\_{\mathfrak{A}}(P\_3) = \begin{bmatrix} \frac{3}{2}\alpha & 1-\alpha & \frac{1}{2}(1-\alpha) \\ 1-\alpha & 2(1-\alpha) & 1-\alpha \\ \frac{1}{2}(1-\alpha) & 1-\alpha & \frac{3}{2}\alpha \end{bmatrix}.$$

Using the software Maple 18, it is easy to calculate that the generalized reciprocal distance spectrum of the path of order 3 is { <sup>3</sup> <sup>2</sup> *<sup>α</sup>* <sup>+</sup> <sup>1</sup> <sup>4</sup> <sup>+</sup> <sup>1</sup> 4 √ 36*α*<sup>2</sup> − 68*α* + 33, <sup>3</sup> <sup>2</sup> *<sup>α</sup>* <sup>+</sup> <sup>1</sup> <sup>4</sup> <sup>−</sup> <sup>1</sup> 4 √ 36*α*<sup>2</sup> − 68*α* + 33, 2*α* − <sup>1</sup> <sup>2</sup> }, this is false.

Therefore, G does not have two vertices at distance two or more. Then, *G* = *Kn*.

$$\begin{array}{c} \textbf{Lemma 8 ([13])}. \quad \textbf{If } \textbf{x}\_1 \ge \textbf{x}\_2 \ge \cdots \ge \textbf{x}\_m \text{ are real numbers such that } \sum\_{i=1}^m \textbf{x}\_i = \textbf{0}, \text{ then } \\\\ \boxed{\textbf{\underline{\underline{\underline{\underline{\underline{\underline{\underline{\underline{\underline{\underline{\underline{\underline{\underline{\underline{\underline{\underline{\underline{\underline{\alpha}}}}}}}}}}}}}}}} \dots} \end{\underline{\underline{\underline{\underline{\underline{\underline{\underline{\underline{\underline{\underline{\underline{\underline{\alpha}}}}}}}}}}}}} $$

$$\mathbf{x}\_1 \le \sqrt{\frac{m-1}{m} \sum\_{i=1}^m \mathbf{x}\_i^2}.$$

*The equality holds if and only if x*<sup>2</sup> = *x*<sup>3</sup> = ··· = *xm* = − *<sup>x</sup>*<sup>1</sup> *<sup>m</sup>*−<sup>1</sup> *.*

**Lemma 9** (Rayleigh quotient theorem [14])**.** *let M be a real symmetric matrix of order n whose eigenvalues are λ*<sup>1</sup> ≥ *λ*<sup>2</sup> ≥ ... ≥ *λn. Then, for any n-dimensional nonzero column vector x,*

$$
\lambda\_1 \ge \frac{\mathbf{x}^T \mathbf{M} \mathbf{x}}{\mathbf{x}^T \mathbf{x}} \ge \lambda\_n.
$$

**Lemma 10** ([15])**.** *If diam*(*G*) ≤ 2 *and if none of the three graphs F*1*, F*2*, and F*<sup>3</sup> *depicted in Figure 1 are induced subgraphs of G, then diam*(*L*(*G*)) ≤ 2*.*

**Figure 1.** Graphs *F*1, *F*2, *T*<sup>3</sup> in Lemma 10.
