Some bounds on the distance-sum-connectivity matrix.

The distance-sum-connectivity matrix of a graph G is expressed by δ(i) and δ(j) such that i,j∈V . δ(i) and δ(j) are represented by a sum of the distance matrices for i<v and j<v , respectively. The purpose of this paper is to give new inequalities involving the eigenvalues, the graph energy, the graph incidence energy, and the matching energy. So, we have some results in terms of the edges, the vertices, and the degrees.

Introduction

Let G be a simple, finite, connected graphs with the vertex set and the edge set . By 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 Δ and δ, respectively. If any vertices i and j are adjacent, then we use the notation .

The distance-sum-connectivity matrix is defined by the displacement of the vertex degrees and the distance sum. This matrix is denoted by X and represented in [1-18] by

The distance sum is such that D is the distance matrix. The distance-sum-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 such that . We will accept as the spectral radius of the graph , and we will take as for convenience. Basic properties of are , , and . G is a regular graph with order n if and only if [3]. The energy of 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 , such that . We use as for brevity. The incidence energy of a graph is represented by . See [8] and [9].

The number of k-matchings of a graph G is denoted by . The matching polynomial of a graph is described by (see [6]).

The matching energy of a graph G is defined as

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 and the second greatest eigenvalue using the edge number, the vertex number, and the degree. In Sect. 3.2, we focus on the upper bound for energy of and are concerned with the vertex number, the distance matrix, and the determinant of X. In addition, we deal with some results for the incidence energy of , and we find sharp inequalities of . 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 2.1

([12])

Let be a spectral radius and be an irreducible nonnegative matrix with . Then

Lemma 2.2

([4])

If G is a simple, connected graph and is the average degree of the vertices adjacent to , then

Ozeki established Ozeki’s inequality in [16]. This inequality holds some bounds for our graph energy. This inequality is as follows.

Theorem 2.3

(Ozeki’s inequality)

If , , then where , , , and .

Polya–Szego found an interesting inequality in [17]. This inequality is set as follows.

Theorem 2.4

(Polya–Szego inequality)

If for , then where , , , and .

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 , then

Let be positive real numbers for . is defined as follows:

Lemma 2.6

(Maclaurin’s symmetric mean inequality [2])

Let be real nonnegative numbers, then This equality holds if and only if .

Theorem 2.7

Let G be a simple graph. Let zeros of the matching polynomial of this graph be . Then The zeros of the matching polynomial provide the equations and .

Main results

Upper bounds on eigenvalues

A lot of bounds for the eigenvalues have been found. We now establish further bounds for and 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 . In this note, .

Indeed, McClelland’s famous bound is [15] .

We now will give an upper bound for the eigenvalues of .

Theorem 3.1

If G is a simple, connected graph and D is the distance matrix of G, then

Proof

Let be an eigenvector of (). Let one eigencomponent and the other eigencomponent for every k. Let . We know (). If we take the ith equation of this equation, we obtain We can take each ’s as . 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 be eigenvalues of the distance-sum-connectivity matrix X and be an energy of X, then where is the second greatest eigenvalue of X.

is the second largest eigenvalue of X. Firstly, we show that . We know that (). So, . 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 X, we obtain We know that . Hence, Secondly, we will show that .

We know that and . So, . 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 and , then . So, Since and , then  □

Upper and lower bounds on incidence energy

In the sequel of this paper, we expand bounds under the energy of with n, D and .

Theorem 3.3

Let G be a regular graph of order n with m edges. Let be an incidence energy of and be singular values of . Then

Let and be singular values of and , respectively. We will use that .

By Lemma 2.5, So, Since and , we get Since , Hence,  □

Theorem 3.4

Let G be a graph with n nodes and m edges. Let the smallest and the largest positive singular values and of X, respectively, and be a determinant of the distance-sum-connectivity matrix X of G. For , where is the energy of X.

Suppose and , . Apply Theorem 2.3 to show that Thus, it is readily seen that By the Cauchy–Schwarz inequality, we can express that Then it suffices to check that Since and using Theorem 3.1, we obtain  □

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 be a matching energy of G, then where is the zero of its matching polynomial.

Let be the zeros of their matching polynomial. We suppose that , where and , . By Theorem 2.4, we obtain Since , It is easy to see that We can assume that the maximum is and the minimum is . So the bound can be sharpened, that is,  □

Corollary 3.6

Let G be a k-regular graph. Then

Since G is a k-regular graph, we can take and . By Theorem 3.5, Hence,  □

Theorem 3.7

Let G be a connected graph with n vertices and m edges. Then if and only if .

Let us consider and for . Setting that in Lemma 2.6, we get Also, Since and , then We know that . So, The above equality holds if and only if . □

Theorem 3.8

Let G be a connected graph with n vertices and m edges. Then if and only if .

Let us consider and for . By Lemma 2.6, we determine Using the Cauchy–Schwarz inequality, we have It is clear that the above equality gives Thus, it is pointed out that . Since and , then Hence, The above result holds if and only if . □

Conclusions

The main goal of this work is to examine distance-sum-connectivity matrix 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-sum-connectivity matrix of a graph in terms of its energy, its incidence energy, and its matching energy.