#### Abstract

Let be a simple graph of order . The matrix is called the Laplacian matrix of , where and denote the diagonal matrix of vertex degrees and the adjacency matrix of , respectively. Let , be the largest eigenvalue, the second smallest eigenvalue of respectively, and be the largest eigenvalue of . In this paper, we will present sharp upper and lower bounds for and . Moreover, we investigate the relation between and .

#### 1. Introduction

We begin with the preliminaries which are required throughout this paper. Let be a simple graph with vertex set and edge set . The integers and are the order and the size of the graph , respectively. The open neighborhood of vertex is , and the degree of is . Let be the complete graph of order and be the complement of the graph . Let and be the maximum degree and the minimum degree of the vertices of , respectively. The eigenvalues of the adjacency matrix , are denoted by . The matrix , where is the diagonal matrix of vertex degrees, is called the Laplacian matrix of and rarely appears in the literature. The eigenvalues of Laplacian matrix are denoted as . The Laplacian matrix of a graph and its eigenvalues can be used in several areas of mathematical research and have a physical interpretation in various physical and chemical theories. The adjacency matrix of a graph and its eigenvalues were much more investigated in the past than the Laplacian matrix. Many related physical quantities have the same relation to ; also, there are many problems in physics and chemistry where the Laplacian matrices of graphs and their spectra play the central role. Recently, its applications to several difficult problems in graph theory were discovered (see ).

Merris  discussed the Laplacian matrices of graphs. In , some bounds are established for Laplacian eigenvalues of graphs. Taheri et al.  presented some bounds for the largest Laplacian eigenvalue of graphs. Patra et al.  obtained bounds for the Laplacian spectral radius of graphs. In , the authors investigated some bounds for the Laplacian spectral radius of an oriented hypergraph. Chen  established some bounds for .

In this paper, we first present sharp upper and lower bounds for and , and then we investigate the relation between and .

#### 2. Preliminaries

In this section, some fundamental results that are used in this paper are recalled. We begin with the following result, which plays a key role in this section.

Lemma 1. (see ). Let be a graph of order and size . Then,where is the well-known graph invariant called the first Zagreb index .
Favaron and Mahéo  proved the following result:

Lemma 2. (see ). Let be a graph of order . Then,The proof of the next result can be found in .

Lemma 3. Let be a graph of order . Then, if and only if or .
Das in  proved the following lemma.

Lemma 4. Let be a connected graph of order . Then, if and only if .

In , a class of real polynomials Pn(x) = xn + a1xn−1 + a2xn−2 + b3xn−3+ … + bn, denoted as (a1, a2), where a1 and a2 are fixed real numbers, was considered.

Theorem 1. For the roots of an arbitrary polynomial from this class, the following values were introduced:Then upper and lower bounds for the polynomial roots, , were determined in terms of the introduced values

#### 3. Main Results

In this section, we will obtain some sharp upper and lower bounds for and involving the first Zagreb index and order and size of graphs. Moreover, we investigate the relation between and . The first result is an immediate consequence of Theorem 1 and Lemma 1.

Lemma 5. Let be a graph of order and size . Then,Here, we will obtain a lower and an upper bound for the largest Laplacian eigenvalue and the second smallest Laplacian eigenvalue , respectively.

Theorem 2. Let be a graph of order and size . Then,and the equalities hold if and only if or .

Proof. For every fixed number , we can write thatIt is not hard to see that when or , we getHence, we haveSo, we can writeThis is equivalent to or Therefore, we haveHence, by using Lemma 1, we haveBy combining inequalities (15)–(17), we get the following inequality:By inequalities (5) and (6), we haveTherefore, we haveIf the equality in (7) holds, then the inequality in (10) must hold, and hence we have ; thus, by Lemma 3, we have or . Conversely, if or , then it is not difficult to see that the equalities in (7) and (8) hold.
Next, we present an upper bound for spectral radius of the Laplacian matrix.

Theorem 3. Let be a connected graph of order and size . Then,

Proof. Applying Lemma 1, we can writeorBy inequality (23), we haveUsing inequality (24), we getorBy inequality (26), we can writeSolving this inequality leads toFinally, we will describe a relationship between spectral radius () of the Laplacian matrix and the spectral radius () of the adjacency matrix.

Theorem 4. Let be a connected graph of order and size . Then,and the equality holds if and only if .

Proof. By inequality (26) and Lemma 2, we haveNow suppose that the equality holds in (29). Then, all the inequalities in the proof must be equalities.
If the equality holds in (30), then inequality (23) must be equality; in other words,orTherefore, by equality (33), we getHence, by Lemma 4, we get . Conversely, one can easily see that equality holds in (29) when .

#### 4. Conclusion

In this paper, we established some sharp upper and lower bounds for the largest eigenvalue and the second smallest eigenvalues of Laplacian matrix involving the first Zagreb index and order and size of graphs. Moreover, we investigate a relation between the largest eigenvalues of Laplacian matrix and the adjacency matrix.

There are still open and challenging problems for researchers. For example, the problem of matrix, matrix, and so on remains open for further investigation.

#### Data Availability

The data involved in the examples of our manuscript are included within the article.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.