Abstract
Let be a graph of order with vertices labeled as . Let be the degree of the vertex , for . The difference adjacency matrix of is the square matrix of order whose entry is equal to if the vertices and of are adjacent or 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.
1. Introduction
Let be a simple and connected graph with vertex set and edge set . For , denotes the degree of vertex in . If and are adjacent vertices in , the edge connecting them is denoted by . 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 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 index and Randić index of a (molecular) graph 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 is defined as . The eigenvalues of the difference index are denoted by and are said to be the -eigenvalues of . We note that since the matrix of is symmetric, its eigenvalues are real and can be ordered as .
Some of the most popular topological indices are given as follows: the first Zagreb index usually denoted by is defined as follows [3, 4]:whereas the first Zagreb index satisfies the identity
The modified second Zagreb index 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 defined as
The 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 formwhere is a pertinently chosen function with the property . On each of such topological indices, a matrix can be associated, defined as
There 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]:(ii)Albertson matrix [10]:(iii)Geometric-arithmetic matrix [11]:(iv) 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 Randi index of a graph , defined as follows:
We call as the difference matrix associated with the difference between atom-bond connectivity index and Randić index of graph .
This 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.
2. 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 .
Let be a graph of order with edges and be the adjacency difference matrix with respect to a given degree. Suppose thatis the characteristic polynomial of . Thus, in order to find nontrivial solutions to equation (15), one must demand that is not invertible, or equivalently,
Equation (16) is called the characteristic equation. Evaluating the determinant yields an -th order polynomial in , called the characteristic polynomial, which we have denoted above by .
The determinant in equation (16) can be evaluated by the usual methods. It takes the following form:where and are the vertices degree and .
Now, we begin with the following example.
Example 1. The difference matrix of the graph in Figure 1 is
The characteristic polynomial of the maximum degree matrix isand the difference eigenvalues of are .
Here, we compute some of the coefficients in equality (17).
Lemma 1. Let , and be the coefficients in equality (17), then
Proof. (i)By the definition of the polynomial, we get .(ii)The sum of determinants of all principal submatrices of is equal to the trace of implying that(iii) sum of determinants of all the principal submatrices of :where .
We are dealing this part with some results related to the traces of powers of . Recall that we denote . Now, we prove the following lemma that will need to obtain the main results.
Lemma 2. Let be a graph with vertices and difference matrix of and be the eigenvalues of . Then,where indicates summation over all pairs of adjacent vertices .
Proof. By definition, the diagonal elements of are equal to zero. Therefore, the of is zero.
Next, we calculate the matrix . For ,whereas for ,Therefore,Since the diagonal elements of arewe haveWe next calculate . The diagonal elements of areThen, we obtainThis completes the proof.
Here, we recall the following lemma that we need to prove the next lemma.
Lemma 3 (Rayleigh–Ritz [14]). If is a real symmetric matrix with eigenvalues , then for any , ,
Equality holds if and only if is an eigenvector of , corresponding to the largest eigenvalue .
The following result is related with the large eigenvalue .
Lemma 4. Let be a connected graph with vertices. Then, the spectral radius of the difference matrix is bounded from below as
Proof. Let be the difference matrix corresponding to . By Lemma 3, for any vector ,because . Also,Using equations (34) and (35), by Lemma 3, we obtainSince (36) is true for any vector , by putting , we haveThis completes the proof.
Now, we obtain a lower bound for the maximum eigenvalue.
Lemma 5. Let be a graph with n vertex and edges:
Proof. Let be a unit vector, thenIf we put , we getTherefore, by Lemma 3, the proof is now complete.
Here, we obtain an upper bound for that will need to obtain upper and lower bounds for new energy.
Lemma 6. Let be a simple graph with m edges and the redefined first index. Then,
Proof. By Lemma 2, we know thatNow, since for all edges , for all vertices ,This complete the proof.
3. 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.
The energy of a graph is the sum of the absolute values of the eigenvalues of . The motivation for the introduction of this invariant comes from chemistry, where results on were obtained already in the 1940s. The 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]. The energy of different graphs including regular, nonregular, circulant, and random graphs is also under study. The of the graph is defined aswhere , , are the of graph .
The difference energy ( energy) of the graph 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 [16–18] and [10, 19–26]). These modified energies have applications in theoretical organic chemistry [27], image processing [28], and information theory [29].
Example 2. The difference energy of the graph in Figure 1 isWe start with upper bound for difference energy.
Theorem 1. Let be a graph with vertices. Then,
Proof. By Cauchy Schwarz inequality and Lemma 6, we haveHence,This completes the proof.
We need the following lemma to obtain lower and upper bounds of the difference energy involving .
Lemma 7 (see [30]). Let and be real numbers such that and , . Then,where
Here, we obtain lower and upper bounds for new energy.
Theorem 2. Let be a graph with vertices. Then,
Equality holds if and only if .
Proof. For , , , and , inequality (50) becomesTherefore, by Lemma 2 and definition of difference energy, inequality (52) becomesClearly, if , then ; hence, , and also, ; therefore by (51), we have . Conversely, if equality holds on both sides of (51), then equality holds in (52) and (53); therefore, we haveSo by equality on both sides of (54), we get , since , therefore, ; in other words, or ; hence, , then .
Now, we obtain lower bound for new energy involving and .
Theorem 3. Let be a graph with vertices and at least one edge. Then,
Proof. Using Hlder inequality, we havewhich holds for any nonnegative real numbers . Setting and , we obtainsince is nonempty graph; hence, we haveHence, we get the result.
Now, we obtain an upper bound for new energy involving vertices, edges, minimum degree, and index.
Theorem 4. Let be a nonempty graph with vertices. Then,
Proof. Let be the eigenvalues of . By the Cauchy–Schwartz inequality,Hence,Note that the function decreases for . By Lemma 5, we have ; therefore,So , since is nonempty graph which implies thatThis completes the proof.
Similarly, by Lemma 4, we have the following lemma.
Lemma 8. Let be a nonempty graph with vertices:
Here, we obtain an upper bound for new energy involving vertices and .
Theorem 5. Let be a graph with vertices. Then,
Equality holds if and only if .
Proof. Let and be sequences of real numbers and and are nonnegative. Then, the following inequality is valid (see [31]):For , , , inequality (66) becomesTherefore, by Lemma 2, we haveIf , then it is easy to check that the equality in (65) holds. Conversely, if the equality in (65), then equality holds in (66) and (67); therefore, we haveHence, we haveBy using equality (70), we getHence, by equality (71), we haveTherefore, .
Now, we obtain a lower bound for new energy involving maximum eigenvalue and .
Theorem 6. Let be a nonempty graph with vertices. Then,
Equality holds if and only if .
Proof. Let be decreasing nonnegative sequences with and a nonnegative sequence, for . Then, the following inequality is valid (see [32] p. 85):For and , , inequality (74) becomesSincethen the above inequality follows directly from the assertion of Theorem 6, i.e., inequality (73).
If , then it is easy to check that the equality in (73) holds. Conversely, if the equality in (73) hold, then equality also holds in (74) and (75); therefore, we haveFrom equality (77), we getBy equality (78), we haveEquality (79) implies thatSince is a nonempty graph, then ; therefore, by equality (80), we haveHence, we have . Therefore, .
Analogy with the paper [24], we obtain the following theorems.
Theorem 7. Let be a graph with vertices. Then,
Theorem 8. Let be a connected graph with vertices. Then,
4. Conclusion
In this paper, the spectral properties of the difference between the atom-bond connectivity index and Randić index were studied. Our main tool was the modification of the classical adjacency matrix involving the degree of the vertices in graph . Some properties of its characteristic polynomial were established. In addition, bounds for the difference energy were obtained, as well as the relations between them.
Data Availability
No data were used in this research.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Authors’ Contributions
All authors contributed equally to this work.
Acknowledgments
The research of the second author was supported by the Research Intensified Grant Scheme (RIGS), Phase 1/2019, Universiti Malaysia Terengganu, Malaysia, with Grant Vot. 55192/6