Some bounds on the distance-sum-connectivity matrix

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Introduction
Let G be a simple, finite, connected graphs with the vertex set V (G) and the edge set E(G). By d i we denote the degree of a vertex. Throughout this paper, the vertex degrees are assumed to be ordered non-increasingly. The maximum and the minimum vertex degrees in a graph are denoted by and δ, respectively. If any vertices i and j are adjacent, then we use the notation i ∼ j.
The distance sum is δ(i) = v j=1 ( v D ij ) such that D is the distance matrix. The distancesum-connectivity matrix is an interesting matrix, and this paper deals with bounds of this matrix. We find some upper bounds and these bounds contain the edge numbers, the vertex numbers, and the eigenvalues. The eigenvalues of this matrix are λ 1 ( δ X), λ 2 ( δ X), . . . , λ n ( δ X) such that λ 1 ( δ X) ≥ λ 2 ( δ X) ≥ · · · ≥ λ n ( δ X). We will accept λ 1 ( δ X) as the spectral radius of the graph δ X(G), and we will take λ 1 ( δ X) as λ 1 for convenience.
G is a regular graph with order n if and only if λ 1 ≥ 2m n [3]. The energy of ( δ X) is described as Some properties about the graph energy may be found [7,10]. The incidence energy IE of G is introduced by Joojondeh et al. [13] as the sum of singular values of the incidence matrix of G. The incidence matrix of a graph G is defined as The singular values are q 1 ( δ X), q 2 ( δ X), . . . , q n ( δ X) such that q 1 ( δ X) ≥ q 2 ( δ X) ≥ · · · ≥ q n ( δ X). We use q i ( δ X) as q i for brevity. The incidence energy of a graph is represented by IE = IE(G) = n i=1 q i (G). See [8] and [9]. The number of k-matchings of a graph G is denoted by m(G, k). The matching polynomial of a graph is described by α(G) = α(G, λ) = k≥0 (-1) k m(G, k)λ n-2k (see [6]).
The matching energy of a graph G is defined as [11]).
The paper is planned as follows. In Sect. 2, we explain previous works. In the next section, we give a survey on upper bound for the first greatest eigenvalue λ 1 and the second greatest eigenvalue λ 2 using the edge number, the vertex number, and the degree. In Sect. 3.2, we focus on the upper bound for energy of δ X(G) and are concerned with the vertex number, the distance matrix, and the determinant of δ X. In addition, we deal with some results for the incidence energy of δ X(G), and we find sharp inequalities of IE( δ X(G)). In Sect. 3.3, we determine different results for the matching energy of a graph with some fixed parameters.

Preliminaries
In order to achieve our plan, we need the following lemmas and theorems.

Lemma 2.2 ([4])
If G is a simple, connected graph and m i is the average degree of the vertices adjacent to v i ∈ V , then Ozeki established Ozeki's inequality in [16]. This inequality holds some bounds for our graph energy. This inequality is as follows.
Polya-Szego found an interesting inequality in [17]. This inequality is set as follows.
Let G be a simple graph and X and Y be any real symmetric matrices of G. Let us consider eigenvalues of these matrices. These eigenvalues hold in the following lemma.

Lemma 2.5 ([5]) Let M and N be two real symmetric matrices and
. , x s be positive real numbers for 1 ≤ t ≤ s. M t is defined as follows: Lemma 2.6 (Maclaurin's symmetric mean inequality [2]) Let x 1 , x 2 , . . . , x s be real nonnegative numbers, then

This equality holds if and only
Theorem 2.7 Let G be a simple graph. Let zeros of the matching polynomial of this graph be μ 1 , μ 2 , . . . , μ n . Then The zeros of the matching polynomial provide the equations n i=1 μ 2 i = 2m and i<j μ i μ j = -m.

Upper bounds on eigenvalues
A lot of bounds for the eigenvalues have been found. We now establish further bounds for λ 1 and λ 2 involving the n, m and d. Firstly we give some known bounds about graph theory.
In the reference [14] a lower bound is given: if and only if G consists of a complete bipartite graph K x,y . In this note, xy = m.
. We now will give an upper bound for the eigenvalues of δ X(G).

Theorem 3.1 If G is a simple, connected graph and D is the distance matrix of G, then
If we take the ith equation of this equation, we obtain We can take each D ij 's as D in . So, Using the Cauchy-Schwarz inequality, From Lemmas 2.1 and 2.2, we have Theorem 3.2 Let G be a simple, connected graph with m edges and n vertices. Let λ 1 , λ 2 , . . . , λ n be eigenvalues of the distance-sum-connectivity matrix δ X and E(G) be an energy of δ X, then where λ 2 is the second greatest eigenvalue of δ X.
Proof λ 2 is the second largest eigenvalue of δ X. Firstly, we show that λ 1 ≥ 2m x k . Similar to Theorem 3.1, if we take the ith equation of this equation, we obtain Using the Cauchy-Schwarz inequality and calculating the distance matrices of δ X, we obtain We know that n k=1 = 2m. Hence, Secondly, we will show that λ 2 (G) ≤ 2m( We know that n i=1 λ i = 0 and n i=1 (λ i ) 2 = 2m. So, λ 1 + λ 2 = -n i=3 λ i . Hence, If we take the square of both sides, we obtain By the Cauchy-Schwarz inequality with the above result, we have If we make necessary calculations, we have Since λ 1 ≥ λ 2 and λ 1 ≤ d 1 , then d 1 ≥ λ 1 ≥ λ 2 . So, Since E(G) ≤ √ 2mn and λ 1 ≥ 2m , then

Upper and lower bounds on incidence energy
In the sequel of this paper, we expand bounds under the energy of δ X(G) with n, D and det( δ X(G)).

Theorem 3.4 Let G be a graph with n nodes and m edges. Let the smallest and the largest
positive singular values σ 1 and σ n of δ X, respectively, and det( δ X) be a determinant of the distance-sum-connectivity matrix δ X of G. For n > 1, where E(G) is the energy of δ X.
Proof Suppose a i = 1 and b i = σ i , 1 ≤ i ≤ n. Apply Theorem 2.3 to show that (3.14) Thus, it is readily seen that By the Cauchy-Schwarz inequality, we can express that Then it suffices to check that Since λ 1 ≥ λ 2 ≥ · · · ≥ λ n and using Theorem 3.1, we obtain E(G) ≤ n 2 2(n -1)

Upper and lower bounds for matching energy
We determine an upper bound for the matching energy applying the Polya-Szego inequality, and we give some results using Maclaurin's symmetric mean inequality.

Theorem 3.5 Let G be a connected graph and ME(G) be a matching energy of G, then
where μ i is the zero of its matching polynomial.
Proof Let μ 1 , μ 2 , . . . , μ n be the zeros of their matching polynomial. We suppose that s i = |μ i |, where s 1 ≤ s 2 ≤ · · · ≤ s n and t i = 1, 1 ≤ i ≤ n. By Theorem 2.4, we obtain (3.25) It is easy to see that We can assume that the maximum |μ i | is s n and the minimum |μ i | is s 1 . So the bound can be sharpened, that is, if and only if μ 1 = μ 2 = · · · = μ n .

Conclusions
The main goal of this work is to examine distance-sum-connectivity matrix δ X. We find some upper bounds for the distance-sum-connectivity matrix of a graph involving its degrees, its edges, and its eigenvalues. We also give some results for the distance-sumconnectivity matrix of a graph in terms of its energy, its incidence energy, and its matching energy.