General Zagreb adjacency matrix

Let A ( G ) and D ( G ) be the adjacency matrix and the degree diagonal matrix of a graph G , respectively. For any real number α , the general Zagreb adjacency matrix of G is deﬁned as Z α ( G ) = D α ( G )+ A ( G ) . In this paper, the positive semideﬁniteness, spectral moment, coeﬃcients of characteristic polynomials, and energy of the general Zagreb adjacency matrix are studied. The obtained results extend the corresponding results concerning the signless Laplacian matrix, the vertex Zagreb adjacency matrix, and the forgotten adjacency matrix.


Introduction
Let G be a simple graph with the vertex set V (G) and edge set E(G). For v i ∈ V (G), d i or d(v i ) denotes the degree of the vertex v i in G. Recently, in order to extend the spectral theory of classical graph matrices such as adjacency matrix, signless Laplacian matrix and distance matrix, many scholars have devoted themselves to the study of the generalization of graph matrices, and proposed many new graph matrices including the generalised adjacency matrix [4], the universal adjacency matrix [6], A α -matrix [10], and the generalized distance matrix [2]. Inspired by these studies, we propose the general Zagreb adjacency matrix of a graph G as follows: where A(G) and D(G) are the adjacency matrix and the degree diagonal matrix of G, respectively. The general Zagreb adjacency matrix gives several existing matrices as special cases: Let z 1 , z 2 , . . . , z n be the eigenvalues of the general Zagreb adjacency matrix of a graph G with n vertices. The general Zagreb adjacency energy of G is defined as where M α = vi∈V (G) d α i is called the first general Zagreb index [8]. In this paper, some spectral properties of the general Zagreb adjacency matrix are reported. The obtained results extend the corresponding results concerning the signless Laplacian matrix, the vertex Zagreb adjacency matrix, and the forgotten adjacency matrix.

Preliminaries
For an integer k, the k-th spectral moment of a graph is defined as the sum over the k-th powers of all eigenvalues of the adjacency matrix. Let λ i and tr(A) be the ith eigenvalue and trace of the adjacency matrix A, respectively. Denote by P n and C n the path and the cycle, respectively, on n vertices. For a graph G with n vertices and m edges, it holds that where |C 3 | and |C 4 | are the number of triangles and quadrangles of G, respectively. In 1998, Bollobás and Erdős [1] defined the general Randić index as: where α is an arbitrary real number.
Lemma 2.1 (see [11]). Let M = (m ij ) be a matrix with the characteristic polynomial . Then the coefficients of Φ(M ) satisfy the following equations:

The positive semidefiniteness of the general Zagreb adjacency matrix
Theorem 3.1. Let G be a connected graph with n vertices. If α > β, then for k = 1, 2, . . . , n.
Proof. By Weyl's inequality, we have This completes the proof. Proof. It is well known that the signless Laplacian matrix Z 1 (G) is positive semidefinite. If α > 1, and G is a graph with no isolated vertices, then by Theorem 3.1 one has z min (Z α (G)) > z min (Z 1 (G)) ≥ 0.
Proof. Let V 1 , V 2 , . . . , V χ be the color classes of G. For an integer k, 1 ≤ k ≤ χ, define a vector X = (x 1 , x 2 , . . . , x n ) by By the Rayleigh-Ritz theorem, one has On the one hand, for ||X|| 2 it holds that Therefore, Adding the above inequalities for all k ∈ {1, 2, . . . , χ}, one arrives at that is, This completes the proof.
Remark 3.1. Lima et al. [9] showed that if G is a graph with n vertices, m edges and chromatic number χ, then Theorem 3.3 asserts that this bound can be extended to all matrices Z α .

Corollary 3.2.
If M α < 2m χ−1 , and G is a graph, then Z α (G) is not positive semidefinite. Question 3.1. Given a graph G, find the smallest α for which Z α (G) is positive semidefinite.
Since tr(BC) = tr(CB) and tr(D 2α A) = 0, it holds that Since tr(BC) = tr(CB) and tr(D 3α A) = 0, one has This completes the proof.
Proof. From Lemma 2.1 and Theorem 4.1, the results follow. Proof. The result follows from Theorem 4.1.