Spectral Properties with the Difference between Topological Indices in Graphs

Let G be a graph of order n with vertices labeled as v1, v2, . . . , vn. Let di be the degree of the vertex vi, for i � 1, 2, . . . , n. 'e difference adjacency matrix of G is the square matrix of order n whose (i, j) entry is equal to ( ��������� di + dj − 2 􏽱 − 1)/( ���� didj 􏽱 ) if the vertices vi and vj of G are adjacent or (vivj ∈ E(G)) and zero otherwise. Since this index is related to the degree of the vertices of the graph, our main tool will be an appropriate matrix, that is, a modification of the classical adjacency matrix involving the degrees of the vertices. In this paper, some properties of its characteristic polynomial are studied.We also investigate the difference energy of a graph. In addition, we establish some upper and lower bounds for this new energy of graph.


Introduction
Let G be a simple and connected graph with vertex set V(G) � v 1 , v 2 , . . . , v n and edge set E(G). For i � 1, 2, . . . , n, d i denotes the degree of vertex v i in G. If v i and v j are adjacent vertices in G, the edge connecting them is denoted by v i v j . Graph theory has provided the chemists with a variety of useful tools, one of which is the topological indices. A topological index or graph invariant, for a graph, is a numeric quantity which is invariant under isomorphism of the graph. Usage of topological indices in chemistry began in 1947 when chemist Harold Wiener developed the most widely known topological descriptor, the Wiener number (later known as Wiener index), and used it to determine physical properties of types of alkanes known as paraffin [1]. Furthermore, the difference between ABC index and Randić index of a (molecular) graph G was introduced by Ali and Du [2] as follows: By equation (1), the difference between atom-bond connectivity index and Randić index (or difference index) of a graph G is defined as D(G) � ( . e eigenvalues of the difference index D(G) are denoted by ϱ 1 , ϱ 2 , . . . , ϱ n and are said to be the D-eigenvalues of G. We note that since the matrix of D(G) is symmetric, its eigenvalues are real and can be ordered as ϱ 1 ≥ ϱ 2 ≥ · · · ≥ ϱ n .
Some of the most popular topological indices are given as follows: the first Zagreb index usually denoted by M 1 is defined as follows [3,4]: e modified second Zagreb index M * 2 (G) is equal to the sum of the reciprocal products of degrees of pairs of adjacent vertices [5], that is, Ranjini et al. [6] redefined the Zagreb indices, i.e., the redefined first index for a graph G defined as e ABC index was defined as follows [7]: For the study of topological indices of graph associated with group, refer [8,9] and the references therein.
In mathematical chemistry, observe that all these topological indices are of the form where F is a pertinently chosen function with the property F(x, y) � F(y, x). On each of such topological indices, a matrix TI can be associated, defined as ere are several degree-based topological indices introduced to test the properties of compounds and drugs, which have been widely used in chemical and pharmacy engineering. Several matrices which are related to topological indices are given as follows: (i) First Zagreb matrix [4]: (iii) Geometric-arithmetic matrix [11]: (iv) ABC matrix [12].
(v) Sum-connectivity matrix [13]: In this paper, we would like to introduce the matrix associated with the difference between atom-bond connectivity index and Randic � index of a graph G, defined as follows: We call D as the difference matrix associated with the difference between atom-bond connectivity index and Randić index of graph G.
is paper is organized as follows: in Section 2, we introduce some properties of characteristic polynomials of difference matrix and some properties of difference eigenvalues; in Section 3, we study the energy of graphs and introduce the difference energy; and we also obtain lower and upper bounds for this new energy.

Some Properties of Difference Matrix of Graphs
In this section, we introduce some properties of characteristic polynomials of difference matrix and some properties of difference eigenvalues of a graph G. Let G � (V, E) be a graph of order n with m edges and D(G) be the adjacency difference matrix with respect to a given degree. Suppose that P(G, ϱ) :� det(ϱI − D(G)) � a 0 ϱ n + a 1 ϱ n− 1 + a 2 ϱ n− 2 + a 3 ϱ n− 3 + · · · + a n , (15) is the characteristic polynomial of D(G). us, in order to find nontrivial solutions to equation (15), one must demand that D − ϱI is not invertible, or equivalently, Equation (16) is called the characteristic equation. Evaluating the determinant yields an n-th order polynomial in ϱ, called the characteristic polynomial, which we have denoted above by P(G, ϱ). e determinant in equation (16) can be evaluated by the usual methods. It takes the following form: where D � [d ij ] and d ij are the vertices degree v i and v j . Now, we begin with the following example.
Example 1. e difference matrix of the graph G 1 in Figure 1 is e characteristic polynomial of the maximum degree matrix D(G 1 ) is Here, we compute some of the coefficients in equality (17). Let a 0 , a 1 , a 2 , and a 3 be the coefficients in equality (17), then Proof (i) By the definition of the polynomial, we get a 0 � 1. (ii) e sum of determinants of all 1 × 1 principal submatrices of D(G) is equal to the trace of D implying that where P � 1≤i<j<k≤n (( We are dealing this part with some results related to the traces of powers of D. Recall that we denote N k � Trc(D k ). Now, we prove the following lemma that will need to obtain the main results. □ Lemma 2. Let G be a graph with n vertices and difference matrix of D and ϱ 1 , ϱ 2 , . . . , ϱ n be the eigenvalues of D(G). en, where i∼j indicates summation over all pairs of adjacent vertices v i and v j .
Proof. By definition, the diagonal elements of D are equal to zero. erefore, the trace of D is zero. Next, we calculate the matrix D 2 . For i � j, whereas for i ≠ j, erefore, Since the diagonal elements of D 3 are we have We next calculate N 4 . e diagonal elements of D 4 are Journal of Mathematics en, we obtain is completes the proof. Here, we recall the following lemma that we need to prove the next lemma. □ Lemma 3 (Rayleigh-Ritz [14]). If B is a real symmetric n × n matrix with eigenvalues λ 1 (B) ≤ λ 2 (B) ≤ · · · ≤ λ n (B), then for any X ∈ R n , (X ≠ 0), Equality holds if and only if X is an eigenvector of B, corresponding to the largest eigenvalue λ 1 (B). e following result is related with the large eigenvalue ϱ 1 .

Lemma 4.
Let G be a connected graph with n ≥ 2 vertices. en, the spectral radius of the difference matrix is bounded from below as Proof. Let D � ‖d ij ‖ be the difference matrix corresponding to D. By Lemma 3, for any vector X � (x 1 , x 2 , . . . , x n ) t , because d ij � d ji . Also, Using equations (34) and (35), by Lemma 3, we obtain Since (36) is true for any vector X, by putting X � (1, 1, . . . , 1) t , we have is completes the proof. Now, we obtain a lower bound for the maximum eigenvalue. □ Lemma 5. Let G be a graph with n vertex and m edges: Proof. Let x ∈ R n be a unit vector, then If we put (40) erefore, by Lemma 3, the proof is now complete. Here, we obtain an upper bound for N 2 that will need to obtain upper and lower bounds for new energy. □ Lemma 6. Let G be a simple graph with m edges and the redefined first ReZG 1 (G) index. en, Proof. By Lemma 2, we know that is complete the proof.

Bounds for the Difference Energy of Graphs
In this section, we study the energy of graphs and introduce the difference energy. We also obtain lower and upper bounds for the new energy. e energy E � E(G) of a graph G is the sum of the absolute values of the eigenvalues of G. e motivation for the introduction of this invariant comes from chemistry, where results on E were obtained already in the 1940s. e graph energy is a graph-spectrum-based quantity, introduced in the 1970s. After a latent period of 20-30 years, it became a popular topic of research both in mathematical chemistry and in "pure" spectral graph theory. In 1978, Gutman defined energy mathematically for all graphs [15]. e energy of different graphs including regular, nonregular, circulant, and random graphs is also under study. e energy of the graph G is defined as where λ i , i � 1, 2, . . . , n, are the eigenvalues of graph G. e difference energy (E D energy) of the graph G is defined in an analogue way as It is usual and useful to define some modified energies such as Zagreb energy, harmonic energy, Albertson energy, matching energy, Laplacian energy, and geometric-arithmetic energy (refer [16][17][18] and [10,[19][20][21][22][23][24][25][26]). ese modified energies have applications in theoretical organic chemistry [27], image processing [28], and information theory [29].

Example 2.
e difference energy of the graph G 1 in Figure 1 is We start with upper bound for difference energy.
Here, we obtain lower and upper bounds for new energy.

Theorem 2.
Let G be a graph with n vertices. en, Equality holds if and only if G � G.
Theorem 5. Let G be a graph with n vertices. en, Equality holds if and only if G � (n/2)K 2 , (n � 2m).