On the spectral radius and energy of the degree distance matrix of a connected graph

Zia Ullah Khan; Abdul Hameed

doi:10.1515/math-2025-0139

Article Open Access

On the spectral radius and energy of the degree distance matrix of a connected graph

Zia Ullah Khan and Abdul Hameed

Published/Copyright: April 17, 2025

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$Open Mathematics$

From the journal Open Mathematics Volume 23 Issue 1

Abstract

Let G be a simple connected graph on n vertices. The degree of a vertex v ∈ V ( G ) , denoted by d v , is the number of edges incident with v and the distance between any two vertices u , v ∈ V ( G ) , denoted by d u v , is defined as the length of the shortest path from u to v . The distance matrix of G , denoted by, D ( G ) , is defined as D ( G ) = d u v . We now define and investigate the degree distance matrix of a connected graph G , defined as M D D ( G ) = ( ( d u + d v ) d u v ) u , v ∈ V ( G ) . In this article, first, we derive bounds for the largest eigenvalue of the degree distance matrix. Then, we establish some bounds for the energy of the degree distance matrix of G .

Keywords: graph; matrix; distance matrix; spectrum; eigenvalue

MSC 2010: 05C50; 05C12; 05C92

1 Introduction

Throughout this study graphs are assumed to be finite, simple, connected and undirected. Let G = ( V ( G ) , E ( G ) ) be an undirected simple graph with vertex set V ( G ) = { v 1 , v 2 , … , v n } and the edge set E ( G ) = { e 1 , e 2 , … , e m } , where ∣ V ( G ) ∣ = n and ∣ E ( G ) ∣ = m are, respectively, called the order and the size of G . We denote the degree of vertex v i , i = 1 , 2 , … , n by d i = d v i . The distance between vertex u and v in G , denoted by d G ( u , v ) or for simplicity d u v , is the length of a shortest path from vertex u to vertex v in G . The maximum distance between any two vertices in G is called diameter of G , denoted by d ( G ) or simply d . Two vertices u , v ∈ V ( G ) are said to be adjacent if they share the same edge, i.e., u ∼ v . For u ∈ V ( G ) the number of vertices adjacent to u is defined as the degree of vertex u and denoted as d u . A graph G is said to be k -regular graph, if each vertex of G has degree k . A root vertex is a vertex u with a directed path from u to v for every vertex u ≠ v in the graph. A graph G is called bipartite if its vertex set V can be partitioned into two disjoint subsets, say V 1 and V 2 such that all edges of G meet both V 1 and V 2 . The distance matrix of G , denoted by D ( G ) , is defined as D ( G ) = ( d G ( u , v ) = d u v ) u , v ∈ V ( G ) . The distance spectral radius μ ( G ) is the largest eigenvalue of the distance matrix D ( G ) .

The study of the distance matrix is of great significance and interest. Consonni and Todeschini [1] briefly studied and pointed out that distance spectral radius is widely applicable as a molecular descriptor in QSPR modeling. The distance eigenvalues and especially the distance spectral radius have been extensively studied for few decades, refer the recent survey [2] along with references therein [3–7]. Dobrynin and Kochetova [8] introduced degree and distance-based molecular index for graphs called degree distance index. In chemical graph theory literature, degree distance index D I was investigated by Tomescu [9,10], Bucicovschi and Cioabǎ [11]. For any i , j ∈ V ( G ) , the degree distance index of a connected graph G is defined as,

D I ( G ) = 2 ∑ i < j ( d i + d j ) d i j .

Indices are commonly used in mathematical chemistry since graphs often represent ideal atomic or molecular structures. Ivanciuc [12] explored the distance valency matrices and introduced molecular graph descriptors, defining the degree distance valency matrix. These matrices effectively represent chemical structures numerically and are useful for analyzing structural properties and molecular studies. Using the molecular graphs, one can easily determine the structure of various chemical compounds by means of its spectral moment, spectra, polynomials, walks, and paths. For more details on some different molecular indices and their applications, readers are suggested to refer [13–15].

Throughout the study, we denote the complete graph, the star graph, and the complete bipartite graph on vertices by K n , S n , and K a , b ( a + b = n ) , respectively. For other concepts and results about graphs not mentioned here, refer, e.g., Diestel [16].

In this work, using the concept of degree distance index [8] and valency matrices [12], we investigate the degree distance matrix. Let G be a connected graph of order n . We denote the degree distance matrix of G by M D D ( G ) . It is the n × n symmetric matrix whose entries m i j are

(1) m i j = ( d i + d j ) d i j , if i ≠ j , 0 , otherwise .

As G is connected, it is obvious from the definition of the degree distance matrix that m i j ⩾ 2 for i ≠ j and m i i = 0 . Let λ 1 ⩾ λ 2 ⩾ … ⩾ λ n be the non-increasing eigenvalues of M D D ( G ) , where the largest eigenvalue λ 1 is called the degree distance spectral radius of G , and will be denoted by ρ ( G ) . The spectrum is the set of all the eigenvalues of the matrix M D D ( G ) denoted by σ M D D ( G ) . Let D i be the sum of the i th row or column of M D D ( G ) , then clearly

D I ( G ) = ∑ i = 1 n D i .

For illustration of the degree distance matrix, we present the following example.

Example 1.1

Let us consider a wheel graph W 6 on six vertices depicted as follows (Figure 1):

The degree distance spectrum of W 6 is the following one:

σ M D D ( W 6 ) = λ 1 λ 2 λ 3 λ 4 λ 5 λ 6 43.37 − 2.29 − 2.29 − 7.37 − 15.70 − 15.70 .

$Figure 1 Wheel graph W 6 {W}_{6} and its corresponding M D D {M}_{DD} matrix.$

Figure 1

Wheel graph W 6 and its corresponding M D D matrix.

In addition to the above example, one can easily calculate the degree distance spectrum for a complete graph K n whose M D D matrix is as follows:

M D D ( K n ) = 2 ( n − 1 ) ( J − I ) ,

where J n × n is the matrix of all one’s and I n × n is the identity matrix. Thus, its spectrum is of the following form:

(2) σ M D D ( K n ) = { ( − 2 ( n − 1 ) ) ( n − 1 ) , 2 ( n − 1 ) 2 } .

Note that n − 1 and 2 in exponent indicates the multiplicity of the corresponding eigenvalue.

The energy of a graph G is defined to be the sum of the absolute values of the eigenvalues of its adjacency matrix, denoted by E ( G ) . In graph theory, the energy for any graph G was established by Gutman [17] in 1978. However, the base for graph energy definition appeared much earlier in 1930s from the molecular orbital theory, which allows chemists to approximate energies associated with π -electron orbitals. In graph theory, the energies for graphs are defined according to their corresponding matrices. For brief study on different types of graph energies, the readers may refer to [17,18,19,20].

Let μ 1 , μ 2 , … , μ n be the eigenvalues of the distance matrix of a connected graph G . Indulal et al. [20] defined the distance energy of a graph G as the sum of the absolute values of its distance eigenvalues, i.e.,

E D ( G ) = ∑ i = 1 n ∣ μ i ∣ .

In a similar fashion, we introduce the M D D energy of a graph denoted by E ( G ) , which is equal to the sum of absolute values of the eigenvalues of degree distance matrix,

(3) E ( G ) = ∑ i = 1 n ∣ λ i ∣ ,

where λ 1 , λ 2 , … , λ n are the degree distance eigenvalues of G . For instance, using Equations (2) and (3), the degree distance energy for a complete graph K n is of the following form:

E ( G ) = ( n − 1 ) ∣ − 2 ( n − 1 ) ∣ + ∣ 2 ( n − 1 ) 2 ∣ = 4 ( n − 1 ) 2 .

The rest of the study is organized as follows: In Section 2, we give some basic properties of M D D ( G ) and the degree distance spectral radius of graphs that will be used in the later sections of the study. We also prove some basic results. In Section 3, we discuss the upper and lower bounds for the spectral radius of the M D D ( G ) . In Section 4, we discuss the degree distance energy of connected graphs and propose some sharp upper and lower bounds for the degree distance energy of a graph G . In Section 5, we give a brief summary of this work.

2 Basic properties of degree distance matrix

In this section, we review some definitions and results relevant to our work. We prove some auxiliary results.

Let G be a connected graph with V ( G ) = { v 1 , … , v n } . A column vector x = ( x 1 , … , x n ) ⊤ ∈ R n can be considered as a function defined on V ( G ) , which maps vertex v i to x i , for i = 1 , … , n . Then,

x T M D D ( G ) x = ∑ i = 1 n ∑ j = 1 n ( d i + d j ) d i j x i x j ,

or equivalently,

x T M D D ( G ) x = 2 ∑ i < j ( d i + d j ) d i j x i x j .

Here, we recall that a matrix is irreducible because of the connectedness assumption. Since M D D ( G ) is a non-negative irreducible matrix, by the Perron-Frobenius theorem, ρ ( G ) is simple and there is a unique positive unit eigenvector corresponding to ρ ( G ) , which is called the DD-Perron vector of G . If x is the D D -Perron vector of G , then for each i ∈ V ( G ) ,

ρ ( G ) x i = ∑ j = 1 n ( ( d i + d j ) d i j ) x j ,

or equivalently,

ρ ( G ) x i = d i ∑ j = 1 n d i j x j + ∑ j = 1 n d j d i j x j ,

called the degree distance eigenequation ( D D -eigenequation) of G at vertex i . For a unit column vector x ∈ R n with at least one non-negative entry, by Rayleigh’s principle, we have ρ ( G ) ⩾ x ⊤ M D D ( G ) x with equality if and only if x is the D D -Perron vector of G .

Since M D D ( G ) is a non-negative irreducible matrix, by the Perron-Frobenius theorem, we have the following properties:

Lemma 2.1

Let G be a connected graph of order n. Let x be an eigenvector corresponding to ρ ( G ) . Then

ρ ( G ) is the maximum eigenvalue with multiplicity 1;
x is positive and unique up to scaling;
if G * is a spanning subgraph of G, then
ρ ( G * ) ⩽ ρ ( G ) .

The following auxiliary fact can be proved easily.

Lemma 2.2

Let G be a graph of order n and let λ 1 , λ 2 , … , λ n be its degree distance eigenvalues. Then,

∑ i = 1 n λ i = 0 ;
∑ i = 1 n λ i 2 = 2 S , where S = ∑ i < j ( ( d i + d j ) d i j ) 2 .

Two vertices u and v are equivalent in a connected graph G if there exists an automorphism σ : G → G , such that σ ( u ) = v . In the following proposition, we use the vertex equivalence property to obtain an important property of eigenvectors corresponding to ρ ( G ) .

Proposition 2.3

Let u , v ∈ V ( G ) , where G is a connected graph of order n. Also, let x = ( x 1 , x 2 , … , x n ) T be an eigenvector corresponding to ρ ( G ) . If vertices u and v are equivalent in G, then x u = x v .

Proof

Since u , v ∈ V ( G ) and vertices u and v are equivalent in G where G is connected, then there exists a permutation matrix P such that

P − 1 M D D ( G ) P = M D D ( G ) .

Thus, P − 1 M D D ( G ) P x = ρ ( M D D ( G ) ) x . It follows that P x is also an eigenvector of ρ ( M D D ( G ) ) , which is a contradiction to Lemma 2.1 that x is unique. Consequently, P x = x and x u = x v .□

Recall, that for v ∈ V ( G ) , a connected graph G is said to be k -regular, if d v = k for each v ∈ V ( G ) . So, if D ( G ) is the distance matrix of G , then one can see that for a k -regular graph, the degree distance matrix can be written as M D D ( G ) = 2 k D ( G ) . Thus, we have the following lemma.

Lemma 2.4

Let G be a connected k-regular graph on n vertices. Let D ( G ) be its distance matrix with eigenvalues μ 1 ⩾ μ 2 ⩾ … ⩾ μ n . Then, the degree distance spectrum of G is 2 k μ i , where i = 1 , 2 , … , n .

3 Bounds for the degree distance spectral radius

In this section, we discuss bounds on the spectral radius for a connected graph G . We begin with the following auxiliary fact.

Lemma 3.1

Let G be a connected graph of order n. Also, let D I ( G ) be the degree distance index of G. Then,

ρ ( G ) ⩾ 1 n D I ( G ) .

Here the equality holds if row sums are equal.

Proof

Let x = 1 n ( 1, 1 , … , 1 ) T ∈ R n be a unit vector. Then, by Rayleigh quotient we have

ρ ( G ) ⩾ x T M D D ( G ) x = 2 n ∑ i < j ( d i + d j ) d i j = 1 n D I ( G )

with equality only if x is the principal eigenvector of M D D ( G ) , i.e., the row sum is constant for each row in M D D ( G ) .□

In the next result, we establish another bound for ρ ( G ) .

Theorem 3.2

Let G be a connected graph with bipartition V ( G ) = P ∪ Q , where ∣ P ∣ = p , ∣ Q ∣ = q , and p + q = n . Let δ P and δ Q be the minimum degree among vertices V ( P ) and V ( Q ) , respectively. Then,

(4) ρ ( G ) ⩾ 2 γ − 4 γ 2 + 16 δ P δ Q ( n − 1 − p q ) − p q ( δ P + δ Q ) 2 ,

where γ = δ P ( p − 1 ) + δ Q ( q − 1 ) , with equality if and only if G is a complete bipartite graph.

Proof

Let G be a bipartite graph with vertices partition sets P = { 1 , 2 , … , p } and Q = { p + 1 , p + 2 , … , p + q } , where p + q = n . Let x = ( x 1 , x 2 , … , x n ) T be an eigenvector corresponding to the maximum eigenvalue ρ ( G ) such that

x i = min k ∈ P x k and x j = min k ∈ Q x k .

For the component x i , from the eigenvalue equation, we have

ρ ( G ) x i = d i ∑ k = 1 p d i k x k + ∑ k = 1 p d k d i k x k + d i ∑ k = p + 1 p + q d i k x k + ∑ k = p + 1 p + q d k d i k x k , ⩾ 2 ( p − 1 ) δ P x i + 2 ( p − 1 ) δ P x i + q δ P x j + q δ Q x j .

Thus, we have

(5) ρ ( G ) x i ⩾ 4 δ P ( p − 1 ) x i + q ( δ P + δ Q ) x j .

Similarly, for the component x j , we have

ρ ( G ) x j = d j ∑ k = 1 p d j k x k + ∑ k = 1 p d k d j k x k + d j ∑ k = p + 1 p + q d j k x k + ∑ k = p + 1 p + q d k d j k x k

(6) ρ ( G ) x j ⩾ p ( δ P + δ Q ) x i + 4 δ Q ( q − 1 ) x j .

Combining inequalities (5) and (6), we obtain

( ρ ( G ) − 4 δ P ( p − 1 ) ) ( ρ ( G ) − 4 δ Q ( q − 1 ) ) + p q ( δ P + δ Q ) 2 ⩾ 0 .

As p + q = n , so by simplifying the above inequality, we obtain the quadratic form

ρ ( G ) 2 − 4 [ δ P ( p − 1 ) + δ Q ( q − 1 ) ] ρ ( G ) + p q ( δ P + δ Q ) 2 − 16 δ P δ Q ( n − 1 − p q ) ⩾ 0 ,

so (4) follows from the above inequality.

Now for the equality, we have x i = x k for k = 1 , 2 , … , p and x j = x k for k = p + 1 , p + 2 , … , p + q . This means that the Perron eigenvector x corresponding to ρ ( G ) has two different coordinates. If G is not a complete bipartite graph, then q > δ P and p > δ Q . This completes the proof.□

In the next theorem, we give an upper bound for the spectral radius of bipartite graph in terms of maximum degree and diameter d of the graph G .

Theorem 3.3

Let G be a connected bipartite graph with bipartition V ( G ) = P ∪ Q , where ∣ P ∣ = p , ∣ Q ∣ = q , and p + q = n . Let d be the diameter and Δ P and Δ Q be the maximum degree among the vertices from V ( P ) and V ( G ) . Then,

(7) ρ ( G ) ⩽ d [ ξ + ξ 2 + p q ( Δ P − Δ Q ) 2 + 4 Δ P Δ Q ( n − 1 ) ] ,

where ξ = Δ P ( p − 1 ) + Δ Q ( q − 1 ) with equality if and only if G is a complete bipartite graph.

Proof

Let G be a bipartite graph with vertices partition sets P = { 1 , 2 , … , p } and Q = { p + 1 , p + 2 , … , p + q } , where p + q = n . Let x = ( x 1 , x 2 , … , x n ) T be an eigenvector corresponding to maximum eigenvalue ρ ( G ) , such that

x i = max k ∈ P x k and x j = max k ∈ Q x k .

For the component x i from the eigenvalue equation ρ ( G ) x = M D D x , we have

ρ ( G ) x i = d i ∑ k = 1 p d i k x k + ∑ k = 1 p d k d i k x k + d i ∑ k = p + 1 p + q d i k x k + ∑ k = p + 1 p + q d k d i k x k .

So, for component x i , we have eigen equation of the form

(8) ρ ( G ) x i ⩽ 2 d Δ P ( p − 1 ) x i + d q ( Δ P + Δ Q ) x j .

Similarly, for the component x j , we have

ρ ( G ) x j = d j ∑ k = 1 p d j k x k + ∑ k = 1 p d k d j k x k + d j ∑ k = p + 1 p + q d j k x k + ∑ k = p + 1 p + q d k d j k x k .

So,

(9) ρ ( G ) x j ⩽ d p ( Δ P + Δ Q ) x i + 2 d Δ Q ( q − 1 ) x j .

Combining the inequalities (8) and (9), we obtain

( ρ ( G ) − 2 d Δ P ( p − 1 ) ) ( ρ ( G ) − 2 d Δ Q ( q − 1 ) ) ⩽ d 2 p q ( Δ P + Δ Q ) 2 .

From the above inequality, we obtain the quadratic form

ρ ( G ) 2 − 2 d [ Δ P ( p − 1 ) + Δ Q ( q − 1 ) ] ρ ( G ) − d 2 [ p q ( Δ P + Δ Q ) 2 + 4 Δ P Δ Q ( n − 1 ) ] ⩽ 0 .

The result in (7) follows from the above inequality. Now, for the equality, we have x i = x k for k = 1 , 2 , … , p and x j = x k for k = p + 1 , p + 2 , … , p + q . This means that the Perron eigenvector x corresponding to ρ ( G ) has two different coordinates, if G is a complete bipartite graph such that q = Δ P and p = Δ Q .□

In the next theorem, we give the spectral radius for a star graph S n .

Theorem 3.4

Let S n be the star graph on n vertices with rooted vertex i and ρ ( S n ) be the degree distance spectral radius of S n . Then,

ρ ( S n ) = 2 ( n − 2 ) + 4 ( n − 2 ) 2 + n 2 ( n − 1 ) .

Proof

Let S n be the star graph and i be the rooted vertex. The degree distance matrix of the star graph is of the form

M D D ( S n ) = 0 n n … n n 0 4 … 4 n 4 0 … 4 ⋮ ⋮ ⋮ ⋱ ⋮ n 4 4 ⋯ 0 .

Let x be an eigenvector of S n corresponding to eigenvalue ρ ( S n ) . Let x i be the component of x corresponding to the root of center vertex i . As all the n − 1 leaves are similar to each other, by Proposition 2.3, let x j be the component of x corresponding to each leaf of S n . Then, by using the eigenvalue equation ρ ( S n ) x = M D D ( S n ) x , we obtain the system of equations:

(10) ρ ( S n ) x i = n ( n − 1 ) x j ,

(11) ρ ( S n ) x j = n x i + 4 ( n − 2 ) x j .

From equations (10) and (11), eliminating x i and x j , we can obtain a quadratic equation in ρ ( S n ) , with a positive solution.□

4 Energy of the degree distance matrix

In this section, we discuss the energy of the degree distance matrix of a graph. We remind the readers, that the degree distance energy of a graph is equal to the sum of the absolute values of the eigenvalues of degree distance matrix, i.e.,

E ( G ) = ∑ i = 1 n ∣ λ i ∣ .

We approach the study of energy of degree distance matrix by discussing some observations first. As ρ ( G ) is the M D D spectral radius and by Lemma 2.1, ρ ( G ) > 0 . Thus, for any connected graph G on n ⩾ 2 , it follows that

E ( G ) ⩾ 2 ρ ( G ) ,

with equality if and only if G has exactly one positive M D D eigenvalues. From this fact it follows that all the lower bounds for ρ ( G ) can be easily converted to lower bounds for E ( G ) . From Lemma 3.1 we know that ρ ( G ) ⩾ 1 n D I ( G ) and in the above discussion we concluded E ( G ) ⩾ 2 ρ ( G ) , so combining theses two bounds on E ( G ) we state the following proposition.

Proposition 4.1

Let G be a connected graph of order n and D I ( G ) be the degree distance index of G. Then,

E ( G ) ⩾ 2 n D I ( G ) .

Moreover, equality holds if all the row sums are equal.

As an immediate consequences of the above discussion and Lemma 2.4, we see that the degree distance energy coincides with the distance energy. We can state as follows.

Corollary 4.2

Let G be a connected k-regular graph on n vertices. Let D be the distance matrix of G with eigenvalues μ 1 ⩾ μ 2 ⩾ … ⩾ μ n and distance energy E D ( G ) = ∑ i = 1 n ∣ μ i ∣ . Then,

E ( G ) = 2 k E D ( G ) .

In the following results, we will discuss and prove upper and lower bounds for the energy of the degree distance matrix. We recall from Lemma 2.2 that ∑ i = 1 n λ i = 0 and ∑ i = 1 n λ i 2 = 2 S , where S = ∑ i < j ( ( d i + d j ) d i j ) 2 .

Theorem 4.3

Let G be a connected graph on n vertices. Then,

(12) 2 S ⩽ E ( G ) ⩽ 2 n S .

Proof

Let G be a connected graph of order n . We prove the upper bound for equation (12) simply by using Cauchy Schwarz inequality. From Cauchy Schwarz inequality, we have

∑ i = 1 n a i b i 2 ⩽ ∑ i = 1 n a i 2 ∑ i = 1 n b i 2 ,

substituting a i = 1 and b i = λ i

[ E ( G ) ] 2 = ∑ i = 1 n λ i 2 ⩽ n ∑ i = 1 n λ i 2 = n 2 ∑ i < j ( ( d i + d j ) d i j ) 2 = 2 n S ,

so, we obtain our upper bound E ( G ) ⩽ 2 n S . Using Lemma 2.2, the lower bound can be obtained as follows:

□ [ E ( G ) ] 2 = ∑ i = 1 n λ i 2 ⩾ ∑ i = 1 n λ i 2 = 2 S .

Ramane et al. [21] studied some bounds on the energy of distance matrix for connected graphs, which are helpful in generalizing some known bounds especially for the graphs whose diameter does not exceed by 2. Motivated from the idea in [21], and proofs of their distance energy bounds, we also introduce some bounds for the degree distance energy of connected graphs.

Theorem 4.4

Let G be a connected graph of order n. Then,

(14) 2 S + n ( n − 1 ) ∏ i = 1 n ∣ λ i ∣ 2 n ⩽ E ( G ) ⩽ 2 n S .

Proof

First, we consider the upper bound for equation (14). From Theorem 4.3, the upper bound already holds but we will give an alternate proof for upper bound. Let ∑ i = 1 n ∑ j = 1 n ( ∣ λ i ∣ − ∣ λ j ∣ ) 2 be a quantity and its obvious from the quantity that its non-negative. So,

∑ i = 1 n ∑ j = 1 n ( ∣ λ i ∣ − ∣ λ j ∣ ) 2 = n ∑ i = 1 n ∣ λ i ∣ 2 + n ∑ j = 1 n ∣ λ j ∣ 2 − 2 ∑ i = 1 n ∣ λ i ∣ ∑ j = 1 n ∣ λ j ∣ .

By substitution, we obtain

2 n S + 2 n S − 2 E ( G ) 2 ⩾ 0 2 n S ⩾ E ( G ) ,

which is the required upper bound. Now, for the lower bound, the proof is partially similar to that of Theorem 1 in [21]. Using the definition of degree distance energy,

E ( G ) 2 = ∑ i = 1 n ∣ λ i ∣ 2 = ∑ i < j ( λ i ) 2 + 2 ∑ i < j ∣ λ i ∣ ∣ λ j ∣ = 2 S + ∑ i ≠ j ∣ λ i ∣ ∣ λ j ∣ .

As we know that for non-negative numbers, the arithmetic mean is not smaller than the geometric mean,

1 n ( n − 1 ) ∑ i ≠ j ∣ λ i ∣ ∣ λ j ∣ ⩾ ∏ i ≠ j ∣ λ i ∣ ∣ λ j ∣ 2 = ∏ i = 1 ∣ λ i ∣ 2 ( n − 1 ) 2 = ∏ i = 1 n ∣ λ i ∣ 2 n .

So, we have

∑ i ≠ j ∣ λ i ∣ ∣ λ j ∣ ⩾ n ( n − 1 ) ∏ i = 1 n ∣ λ i ∣ 2 n .

Thus, we obtain

E ( G ) 2 ⩾ 2 S + n ( n − 1 ) ∏ i = 1 n ∣ λ i ∣ 2 n .

Hence, the lower bound in equation (13) holds.□

In the next theorem, we find the upper bound for degree distance energy in terms of its spectral radius. As ρ ( G ) is the degree distance spectral radius, we state that next theorem as follows.

Theorem 4.5

Let G be a connected graph of order n. Then,

(14) E ( G ) ⩽ ρ ( G ) + ( n − 1 ) ( 2 S − ρ ( G ) ) .

Proof

Using the definition of degree distance energy,

E ( G ) = ∑ i = 2 n ∣ λ i ∣ + ρ ( G ) , ( E ( G ) − ρ ( G ) ) 2 = ∑ i = 2 n ∣ λ i ∣ 2 .

By applying the Cauchy-Schwarz inequality to the right hand side of the above equation, we have

∑ i = 2 ∣ λ i ∣ 2 ⩽ ( n − 1 ) ∑ i = 2 n ( λ i ) 2 .

Thus,

( E ( G ) − ρ ( G ) ) 2 ⩽ ( n − 1 ) ∑ i = 2 n ( λ i ) 2 .

As from Lemma 2.2, we know that ∑ i = 2 n ( λ i ) 2 = 2 S − ρ ( G ) and by substituting in the above inequality, we obtain our required upper bound of equation (14), i.e.,

E ( G ) ⩽ ρ ( G ) + ( n − 1 ) ( 2 S − ρ ( G ) ) .

This completes the proof.□

5 Conclusion

Inspired by the work of Dobrynin and Kochetova [8] on the distance matrix and Ivanciuc [12] on valency matrices, in this article, we introduced a new concept, the degree distance matrix of a graph, which merges the distance matrix and the degree distance-based topological indices. This matrix has many properties similar to that of a distance matrix. We give several bounds for the degree distance spectral radius and degree distance energy of a connected graph G . Additionally, since the degree distance matrix is a new concept, there are still many open problems for further exploration, especially in other families of graphs.

Acknowledgements

The authors thank the reviewers for their careful reading and comments, which improved the manuscript.

Funding information: The work of Zia Ullah Khan was supported by the National Natural Science Foundation of China (NSFC-RFIS) under grant no. 1231101292.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript. All authors were involved throughout the preparation of the manuscript and contributed to the final version.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: All the used is already included in the proofs and paper. All the data is available in manuscript.

References

[1] V. Consonni and R. Todeschini, New spectral indices for molecule description, MATCH Commun. Math. Comput. Chem. 1 (2008), 314. Search in Google Scholar

[2] M. Aouchiche and P. Hansen, Distance spectra of graphs: A survey, Linear Algebra Appl. 458 (2014), 301–386. 10.1016/j.laa.2014.06.010Search in Google Scholar

[3] Y. J. Lei and X. D. Zhang, Complete multipartite graphs are determined by their distance spectra, Linear Algebra Appl. 448 (2013), 285–291. 10.1016/j.laa.2014.01.029Search in Google Scholar

[4] G. Yu, H. Jia, H. Zhang, and J. Shu, Some graft transformations and its applications on the distance spectral radius of a graph, Appl. Math. Lett. 25 (2012), no. 3, 315–319. 10.1016/j.aml.2011.09.007Search in Google Scholar

[5] D. M. Cvetković, M. Doob, and H. Sachs, Spectra of Graphs, 3rd edition, Johann Ambrosius Barth Verlag, Heidelberg-Leipzig, 1995. Search in Google Scholar

[6] Y. Wang and B. Zhou, On distance spectral radius of graphs, Linear Algebra Appl. 438 (2013), no. 8, 3490–3503. 10.1016/j.laa.2012.12.024Search in Google Scholar

[7] Z. U. Khan and X. D. Zhang, On Dα spectrum of connected graphs, Contrib. Discrete Math. 17 (2022), no. 2, 77–95. 10.55016/ojs/cdm.v17i2.69706Search in Google Scholar

[8] A. A. Dobrynin and A. A. Kochetova, Degree distance of a graph: a degree analogue of the Wiener index, J. Chem. Inf. Comput. Sci. 34 (1994), 1082–1086. 10.1021/ci00021a008Search in Google Scholar

[9] I. Tomescu, Some extremal properties of the degree distance of a graph, Discrete Appl. Math. 98 (1999), 159–163. 10.1016/S0166-218X(99)00117-1Search in Google Scholar

[10] A. I. Tomescu, Unicyclic and bicyclic graphs having minimum degree distance, Discrete Appl. Math. 156 (2008), 125–130. 10.1016/j.dam.2007.09.010Search in Google Scholar

[11] O. Bucicovschi and S. M. Cioabǎ, The minimum degree distance of graphs of given order and size, Discrete Appl. Math. 156 (2008), 3518–3521. 10.1016/j.dam.2008.03.036Search in Google Scholar

[12] O. Ivanciuc, Design of topological indices. Part 11. Distance-valency matrices and derived molecular graph descriptors, Rev. Roum. Chim. 44 (1999), 519–528. Search in Google Scholar

[13] K. Pattabiraman, Degree and distance-based topological indices of graphs, Electron. Notes Discrete Math. 63 (2017), 145–159. 10.1016/j.endm.2017.11.009Search in Google Scholar

[14] K. C Das, I. Gutman, and J. Mohammad, Relations between distance-based and degree-based topological indices, Appl. Math. Comput. 270 (2015), 142–147. 10.1016/j.amc.2015.08.061Search in Google Scholar

[15] I. Gutman, Degree-based topological indices, Croat. Chem. Acta 86 (2013), no. 4, 351–361. 10.5562/cca2294Search in Google Scholar

[16] R. Diestel, Graph Theory, Graduate Texts in Mathematics, vol. 173, 4th edn., Springer, Heilderberg, 2010. 10.1007/978-3-642-14279-6Search in Google Scholar

[17] I. Gutman, The energy of a graph, Ber. Math-Statist. Sekt. Forschungsz. Graz 103 (1978), 1–22. Search in Google Scholar

[18] I. Gutman and D. Vidovic, Quest for molecular graphs with maximal energy: A computer experiment, J. Chem. Inf. Comput. Sci. 41 (2001), no.4, 1002–1005. 10.1021/ci000164zSearch in Google Scholar PubMed

[19] G. Indulal and Vijayakumar, On a pair of equienergetic graphs, MATCH Commun. Math. Comput. Chem. 55 (2006), 83–90. Search in Google Scholar

[20] G. Indulal, I. Gutman, and A. Vijayakumar, On distance energy of graphs, MATCH Commun. Math. Comput. Chem. 60 (2008), 461–472. Search in Google Scholar

[21] H. S. Ramane, D. S. Revankar, I. Gutman, S. B. Rao, B. D. Acharya, and H. B. Walikar, Estimating the distance energy of graphs, Graph Theory Notes New York 55 (2008), 27–32. Search in Google Scholar

Received: 2024-04-26

Revised: 2024-12-19

Accepted: 2025-02-07

Published Online: 2025-04-17

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2025-0139

Keywords for this article

graph; matrix; distance matrix; spectrum; eigenvalue

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