On the Spectrum of Laplacian Matrix

Let G be a simple graph of order n. +e matrixL(G) � D(G) − A(G) is called the Laplacian matrix of G, where D(G) and A(G) denote the diagonal matrix of vertex degrees and the adjacency matrix of G, respectively. Let l1(G), ln− 1(G) be the largest eigenvalue, the second smallest eigenvalue of L(G) respectively, and λ1(G) be the largest eigenvalue of A(G). In this paper, we will present sharp upper and lower bounds for l1(G) and ln− 1(G). Moreover, we investigate the relation between l1(G) and λ1(G).


Introduction
We begin with the preliminaries which are required throughout this paper. Let G be a simple graph with vertex set V � V(G) and edge set E(G). e integers n � n(G) � |V(G)| and ε � ε(G) � |E(G)| are the order and the size of the graph G, respectively.
e open neighborhood of vertex v i is , and the degree of v i is d G (v i ) � d i � |N(v i )|. Let K n be the complete graph of order n and G be the complement of the graph G. Let Δ and δ be the maximum degree and the minimum degree of the vertices of G, respectively. e eigenvalues of the adjacency matrix A(G), are denoted by λ 1 (G) ≥ λ 2 (G) ≥ . . . ≥ λ n (G). e matrix L(G) � D(G) − A(G), where D(G) is the diagonal matrix of vertex degrees, is called the Laplacian matrix of G and rarely appears in the literature. e eigenvalues of Laplacian matrix G are denoted as l 1 (G) ≥ l 2 (G) ≥ . . . ≥ l n (G) � 0. e 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. e 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 L(G); 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 [1][2][3][4][5][6][7]).
Merris [8] discussed the Laplacian matrices of graphs. In [9], some bounds are established for Laplacian eigenvalues of graphs. Taheri et al. [10] presented some bounds for the largest Laplacian eigenvalue of graphs. Patra et al. [11] obtained bounds for the Laplacian spectral radius of graphs. In [12], the authors investigated some bounds for the Laplacian spectral radius of an oriented hypergraph. Chen [13] established some bounds for λ 1 (G).
In this paper, we first present sharp upper and lower bounds for l 1 (G) and l n− 1 (G), and then we investigate the relation between l 1 (G) and λ 1 (G).

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 [14]). Let G be a graph of order n and size ε. en, where M 1 (G) � n i�1 d 2 i is the well-known graph invariant called the first Zagreb index [15].
Favaron and Mahéo [16] proved the following result: Lemma 2 (see [16]). Let G be a graph of order n. en, The proof of the next result can be found in [17].
Das in [18] proved the following lemma.

Lemma 4.
Let G be a connected graph of order n ≥ 3. en, In [14], a class of real polynomials P n (x) � x n + a 1 x n− 1 + a 2 x n− 2 + b 3 x n− 3 + . . . + b n , denoted as P n (a 1 , a 2 ), where a 1 and a 2 are fixed real numbers, was considered.

Theorem 1.
For the roots y 1 ⩾ y 2 ⩾ . . . ⩾ y n of an arbitrary polynomial φ n (y) from this class, the following values were introduced: Then upper and lower bounds for the polynomial roots, y i , i � 1, 2, . . . , n, were determined in terms of the introduced values

Main Results
In this section, we will obtain some sharp upper and lower bounds for l 1 (G) and l n− 1 (G) involving the first Zagreb index and order and size of graphs. Moreover, we investigate the relation between l 1 (G) and λ 1 (G). e first result is an immediate consequence of eorem 1 and Lemma 1.

Lemma 5.
Let G be a graph of order n ≥ 2 and size ε. en, Here, we will obtain a lower and an upper bound for the largest Laplacian eigenvalue l 1 and the second smallest Laplacian eigenvalue l n− 1 , respectively. Theorem 2. Let G be a graph of order n ≥ 3 and size ε. en, and the equalities hold if and only if G � K n or G � K n .
Proof. For every fixed number t, we can write that It is not hard to see that when t � 1 or t � n − 1, we get Hence, we have So, we can write 2 Mathematical Problems in Engineering is is equivalent to or erefore, we have Hence, by using Lemma 1, we have By combining inequalities (15)-(17), we get the following inequality: By inequalities (5) and (6), we have erefore, we have If the equality in (7) holds, then the inequality in (10) must hold, and hence we have l 1 � l 2 � · · · � l n− 1 � 2ε/n − 1; thus, by Lemma 3, we have G � K n or G � K n . Conversely, if G � K n or G � K n , 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 G be a connected graph of order n ≥ 2 and size ε. en, Proof. Applying Lemma 1, we can write or By inequality (23), we have Using inequality (24), we get or By inequality (26), we can write Solving this inequality leads to Finally, we will describe a relationship between spectral radius (l 1 ) of the Laplacian matrix and the spectral radius (λ 1 ) of the adjacency matrix. □ Theorem 4. Let G be a connected graph of order n ≥ 3 and size ε. en, and the equality holds if and only if G � K n .
Proof. By inequality (26) and Lemma 2, we have Mathematical Problems in Engineering Now suppose that the equality holds in (29). en, all the inequalities in the proof must be equalities.
Hence, by Lemma 4, we get G � K n . Conversely, one can easily see that equality holds in (29) when G � K n .

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.
ere are still open and challenging problems for researchers. For example, the problem of ABC matrix, GA matrix, and so on remains open for further investigation.
Data Availability e data involved in the examples of our manuscript are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.