Abstract
The spectral radius of a square matrix is the maximum among absolute values of its eigenvalues. Suppose a square matrix is nonnegative; then, by Perron–Frobenius theory, it will be one among its eigenvalues. In this paper, Perron–Frobenius theory for adjacency matrix of graph with self-loops will be explored. Specifically, it discusses the nontrivial existence of Perron–Frobenius eigenvalue and eigenvector pair in the matrix , where denotes the number of self-loops. Also, Koolen–Moulton type bound for the energy of graph is explored. In addition, the existence of a graph with self-loops for every odd energy is proved.
1. Introduction
Let be a simple undirected graph of order and size . Let be its adjacency matrix with eigenvalues denoted by . Then, the energy [1] is defined as follows:Let be a graph with self-loops obtained from by adding self-loops to each vertices in and . Let represent the degree of vertex in the graph . The adjacency matrix is the square matrix of order whose element is defined as follows:Let , be the eigenvalues of . Then, the energy [2] of graph defined as follows:Also, from the literature [2], it is observed that eigenvalues of satisfy the following relations:Let , , be the auxiliary eigenvalues of . Then, the following relations are satisfied:where .
Chemically, some graphs are associated with a molecular structure of conjugated hydrocarbons, and the sum of absolute values of the adjacency eigenvalues is the total -electron energy of that molecule [1]. A graph with self-loops is used to represent a hetero-conjugated system [3], in which self-loops are used to distinguish hetero atoms present in the molecule. In the existing literature, studies have predominantly been focused on simple graphs, the matrices characterized by a zero diagonal. However, adding self-loops to a graph alters the zero diagonal entries of the adjacency matrix of a simple graph, which subsequently affects its spectral properties. Motivated by this, we investigate the spectral changes that occur with the inclusion of self-loops. For additional terminologies, one can refer to [4, 5].
The paper is organized as follows: in Section 2, we discuss Perron–Frobenius theory and bounds for the spectral radius of . In Section 3, we present Koolen–Moulton type bound for the energy of the graph with self-loops. In Section 4, we discuss the existence of graphs with self-loops having odd energy.
2. On Perron–Frobenius Theory and Spectral Radius
Perron–Frobenius theory is a fundamental theory in the field of linear algebra. It is mainly focused on the properties of nonnegative square matrices in terms of their eigenvalues and eigenvectors. The spectral radius of a matrix is the maximum among absolute values of its eigenvalues, which is not always an eigenvalue of the matrix. However, it will be an eigenvalue that is more dominant in the case of nonnegative matrices. It is stated in the following lemma:
Lemma 1. If is a nonnegative square matrix, then spectral radius is an eigenvalue of and there is a nonnegative non-zero vector such that (Perron–Frobenius property). Moreover, if the matrix is nonnegative, irreducible, and . Then, is algebraically simple, and the corresponding eigenvector is positive (strong Perron–Frobenius property) [4].
The adjacency matrix of the graph with self-loops being nonnegative assures the existence of Perron–Frobenius property. If a graph is connected, then the corresponding adjacency matrix of the graph with self-loops is irreducible, and it also satisfies strong Perron–Frobenius property, which is of not much interest. The existence of this property is also studied in the case of some nonpositive matrices, and the same can be referred in [6, 7].
It is observed that the auxiliary eigenvalues of matrix , i.e., are the eigenvalues of matrix . It has at least one and at most negative entries in the diagonal, if . It serves as a potential example of a matrix having some negative entries. Moreover, it is of the form , where is any scalar and if is an eigenvalue of then is the eigenvalue of . Suppose be the eigenvector corresponding to eigenvalue , then . We know that has Perron–Frobenius eigenpair say i.e., implying which proves the existence of nonnegative eigenvector in matrix . Note that the resulting eigenvalue need not be a spectral radius. For example, consider cycle with one self-loop with .The eigenvalues of are , inferring . The eigenvalues of are , imply . If we take , which being the average of eigenvalues satisfy , it inherits Perron–Frobenius eigenpair from the original matrix . But in general, this is not true for all graphs with self-loops. For instance, consider a complete bipartite graph with one self-loop to a vertex in the partite set of order ; its eigenvalues are . Then, . This observation leads us to prove Theorem 1.
Lemma 2. Let a square matrix be Hermitian, and be the ordered eigenvalues of A. Then for all [4].
Theorem 1. Let be a connected graph with self-loops. Let be Perron–Frobenius eigenpair of matrix , with . Then, is Perron–Frobenius eigenpair of matrix , if and only if the sum of smallest and spectral radius of is greater than or equal to .
Proof. Let be a connected graph with self-loops. Let be Perron–Frobenius eigenpair of matrix , with . Let be Perron–Frobenius eigenpair of matrix . Let be smallest eigenvalue. Suppose, . Then, Implying , a contradiction to the fact that is spectral radius.
Now, let . To prove the converse, it is enough to show . Suppose, , then there exists an eigenvalue and eigenvector pair , , with , satisfying , which further implies the following:We discuss four possible cases and arrive at a contradiction to our assumption.
Case 1: Suppose . Then, by Equation (7), we write the following:since is spectral radius of .
Case 2: Suppose . Then, by Equation (7), we get , which is again a contradiction.
Case 3: Suppose and . Then, by Equation (7), we get , it implies . By using Lemma 2 we write, , a contradiction to .
Case 4: Suppose and . Then, by Equation (7), we get the following:Therefore, , a contradiction to our assumption. Hence, if , then .
Remark 1. For a disconnected graph, Theorem 1 fails. For example, consider the graph with one self-loop in . Then, . The eigenvalues of are , and . Then, . But the eigenvalues of are . Therefore, .
In 1987, Stanley [8] obtained the bound for the spectral radius of adjacency matrix as follows:The key to its proof lies in adjacency matrix having diagonal zero and its entries in set . We now present a bound for the spectral radius of the graph with self-loops, which has a non-zero diagonal.
Theorem 2. Let be the graph with self-loops and . Then,
, where .
Proof. Let be the graph with self-loops and . Let be Perron–Frobenius eigenpair, with having . Let represent vector having entries as in except co-ordinate of (i.e., ) is replaced by if , if . Let denote row of matrix and be the row sum. Since with having imply which results in the following:Applying Cauchy–Schwarz inequality [4], we get the following:Observe thatNowSince,Then we write,Further simplification results in .
Theorem 3. Let be a connected graph with self-loops, and the sum of smallest and spectral radius of be greater than or equal to . Then,
,
where .
Proof. Let be the connected graph with self-loops, and the sum of the smallest and spectral radius of be greater than or equal to . Let be Perron–Frobenius eigenpair. By Theorem 1, will be Perron–Frobenius eigenpair of . That is, , where is chosen as with having . Let represent vector having entries as in except co-ordinate of (i.e., ) being replaced by if , if . Let denote row of matrix . Since with having imply which results in,Applying Cauchy–Schwarz inequality [4], we get the following:NowBy using a similar argument as in the proof of Theorem 2, we write the following:Further simplification results in,
Remark 2. It is observed that in a few cases, the bound in Theorem 2 is more sharper than the bound in Theorem 3. For example, consider with three self-loops, then the observed spectral radius is . The upper bounds obtained by Theorem 2 and 3 are and , respectively.
In some other cases, the bound in Theorem 3 is more sharper than the bound in Theorem 2. For example, consider cycle with one self-loop, then the observed spectral radius is . The upper bound obtained by Theorems 3 and 2 are and , respectively.
3. Koolen–Moulton Bound for a Graph
In 2001, Koolen and Moulton [9] derived the sharp, well-known bound for graph energy called Koolen–Moulton bound for a graph with as follows:In 2007, Nikiforov [10], using matrix norms generalized Koolen–Moulton type bound for energy of () nonnegative complex matrix, say whose element is represented as , with maximum element and as follows:where is the energy of having , as its singular values (for Hermitian matrix, singular values are just the moduli of its eigenvalues), and
Let be any graph based real symmetric adjacency type matrix having the eigenvalues . Suppose , then the energy of [11] is defined as follows:Let for , be the auxiliary eigenvalues of . Then, Equation (23) is written as follows:
For a real symmetric matrix with non-zero trace, it is observed that being the sum of absolute values of eigenvalues of is not equal to , i.e., to conclude the definition of given by V. Nikiforov in Equation (22), coincides with the graph energies in Equation (23) only when the matrix has trace zero. But there are many graph-based matrices studied in literature [2, 12–14] which has non-zero traces. Motivated by this, we use a few more properties of matrices and discuss Koolen–Moulton type bound for graph energy in the presence of self-loops. More on Koolen–Moulton type bound refers to [10, 15–17].
Remark 3. Let be a graph with vertices, then(1)(2), where .
Lemma 3. Let be matrix and . Then, is an eigenvalue of if and only if is an eigenvalue of [4].
Lemma 4. Let be the auxiliary eigenvalues of with trace k. Then,
Proof. Let be the auxiliary eigenvalues of with trace . Let be the eigenvalues of matrix . By Lemma 3, is an eigenvalue of if and only if is an eigenvalue of , i.e., are the eigenvalues of matrix . Then,
Remark 4. Let and be spectral radii of and , respectively. Suppose ( vector having all its entry ) in Lemma 2, then,(1)(2)
Lemma 5. Let be any real symmetric matrix, . Let be any number satisfying . Then, . Moreover, if is nonnegative matrix with , where is the maximum element of , then,
Proof. Let be any real symmetric matrix, . Let . Consider,Further simplification results in,The function in Equation (28) is monotonically decreasing in the interval . Thus, by using in Equation (28) results in,i.e.,Let be nonnegative with , where being the maximum element of . Then, it is followed by Equation (22).
Theorem 4. Let be the adjacency matrix of graph with . Then, , where (Koolen–Moulton-type bound for E(Gs)).
Proof. Let be a graph with and be a graph with self-loops obtained from by adding self-loops. Since we get . Then, by using Lemma 5, we get
4. On Odd Energy of a Graph with Self-Loops
In 2004, Bapat and Pati [18] proved that the energy of a simple graph is never an odd integer. This motivated us to explore whether a graph with self-loops has odd energy. Interestingly, some graphs with self-loops not only have odd energy, but also, for every odd energy, there exists a graph with self-loops. This significant result is discussed in the following theorem:
Theorem 5. For any odd positive integer , there exists a graph with self-loops on even vertices having energy .
Proof. Let be any odd positive integer. Consider a totally disconnected graph on vertices. Let be a graph obtained from by adding self-loops.
Claim: The totally disconnected graph with self-loops has the energy .
It is observed that eigenvalues of are and with algebraic multiplicity and , respectively. Then, the energy of is given by the following:Therefore, for any odd positive integer , a totally disconnected graph with self-loops is obtained serving the purpose.
Remark 5. For any odd , the existence of graph need not be unique, i.e., for some wise choice of , we can construct totally disconnected graph , having energy .
For the construction of the desired graph, let , where .
Consider a totally disconnected graph on vertices with self-loops. In order to obtain the condition on , we look at the energy of totally disconnected graph ,By rearranging, we get the quadratic equation as . Therefore,Since is an integer, we get must be a perfect square; moreover, even. If , then with which is discussed in the proof of Theorem 5. If is non-zero, then by observation, we construct the following pairs forming a perfect square for a given , refer to Table 1.
Let be the sequence of corresponding to . It is observed that sequence is the difference of adjacent terms of (i.e., ), is an arithmetic progression with first term and common difference . Then, the term of the sequence is given by , which on further simplification results in . For a given , choice of can be , by substituting in Equation (33), we get as and . Thus, for odd energy we find two different graphs on vertices with self-loops and . The different pairs of are observed for given , as in Table 2.
Remark 6. Graphs other than totally disconnected graphs are found having odd energy.
Example 1: The graph has energy 5.
Example 2: The resultant graph obtained from complete graph by removing one edge and adding self-loops to vertices incident with it has energy .
Data Availability
All the data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.