Seidel Matrix Research Articles

Comparative Study on Different Types of Energies

Objectives : The absolute sum of the eigenvalues is the definition of the graph's energy. In addition to discussing their relationship, this study compares the energy of the Adjacency matrix, Laplacian matrix, Signless Laplacian matrix, and Seidel matrix applied to ten distinct kinds of graphs. In this study, a correlation is established between the energy of graphs and the energy of Edge Antimagic graphs. Methods: The technical description of the graph's energy, . An Edge Antimagic graph's energy, identified by (G) = , is the absolute total of its eigenvalues. Findings: The energy was calculated along with its link to the energy of Edge Antimagic matrix, Adjacency matrix, Laplacian matrix, Signless Laplacian matrix, and Seidel matrix. Novelty: Edge antimagic graphs have been applied using the concept of energy. The energy of Edge Antimagic graphs and the energy of graphs were discovered to be related. An analysis was conducted comparing Adjacency energy, Seidel energy, Laplacian energy, and Signless Laplacian energy. Keywords: Laplacian Energy, Spectrum, Seidel Matrix, Signless Laplacian Matrix, Edge Antimagic Graphs.

A short proof of Haemers' conjecture on the Seidel energy of graphs

Haemers' conjecture on the Seidel energy of graphs, states that the energy of the Seidel matrix of any graph of order n is at least 2n−2, where the equality holds for the complete graph. In this paper, we give a short simple proof of this conjecture.

On the eigenvalues and Seidel eigenvalues of chain graphs

In this paper, we primarily focus on the eigenvalues of the adjacency matrix and Seidel matrix of chain graphs, referred to as eigenvalues and Seidel eigenvalues of these graphs, respectively. Firstly, we utilize the characteristic polynomial of the adjacency matrix of a chain graph to construct infinite pairs of non-isomorphic cospectral chain graphs. Next, we determine the inertia of the Seidel matrix of a chain graph and establish an interval that does not contain the Seidel eigenvalues of chain graphs. Lastly, we characterize chain graphs with distinct Seidel eigenvalues.

On the Eigenvalues and Energy of the Seidel and Seidel Laplacian Matrices of Graphs

Let S(Γ) be a Seidel matrix of a graph Γ of order n and let D(Γ) = diag(n − 1 − 2d1, n − 1 − 2d2, …, n − 1 − 2dn) be a diagonal matrix with di denoting the degree of a vertex vi in Γ. The Seidel Laplacian matrix of Γ is defined as SL(Γ) = D(Γ) − S(Γ). In this paper, we obtain an upper bound, and a lower bound on the Seidel Laplacian Estrada index of graphs. Moreover, we find a relation between Seidel energy and Seidel Laplacian energy of graphs. We establish some lower bounds on the Seidel Laplacian energy in terms of different graph parameters. Finally, we present a relation between Seidel Laplacian Estrada index and Seidel Laplacian energy of graphs.

Some relations between energy and Seidel energy of a graph

AbstractThe energy E(G) of a graph G is the sum of the absolute values of eigenvalues of G and the Seidel energy ES(G) is the sum of the absolute values of eigenvalues of the Seidel matrix S of G. In this paper, some relations between the energy and Seidel energy of a graph in terms of different graph parameters are presented. Also, the inertia relations between the graph eigenvalue and Seidel eigenvalue of a graph are given. The results in this paper generalize some of the existing results.

Seidel spectra of some variants of corona operations

The Seidel spectrum of a graph is defined as the multiset of all eigenvalues of its Seidel matrix. Recently, there has been a renewed interest in studying Seidel spectrum of graphs and some achievements have been made in this regard. In this paper, we determine the Seidel spectra of some variants of corona operations, such as edge corona, subdivision-vertex corona, subdivision-vertex neighbourhood corona of two graphs and so on. The corresponding Seidel eigenvectors are also described completely. Applying the obtained results, we give some sufficient and necessary conditions for edge corona, subdivision-vertex corona and subdivision-vertex neighbourhood corona of two graphs to be Seidel integral.

On the Seidel Estrada index of graphs

For a simple graph G on n vertices the Seidel Estrada index of G, denoted by SEE ( G ) , is defined as SEE ( G ) = ∑ i = 1 n e θ i , where θ 1 , … , θ n are the Seidel eigenvalues (the eigenvalues of the Seidel matrix) of G. In this paper, we find the maximum and minimum values of the Seidel Estrada index among all graphs with the fixed number of vertices. Our results confirm some conjectures on Seidel Estrada index of graphs that have been posed in [M. Hakimi-Nezhaad, M. Ghorbani, On the Estrada Index of Seidel Matrix, Mathematics Interdisciplinary Research 5 (2020) 43–54].

Small-World Networks with Unitary Cayley Graphs for Various Energy Generation

Complex networks have been a prominent topic of research for several years, spanning a wide range of fields from mathematics to computer science and also to social and biological sciences. The eigenvalues of the Seidel matrix, Seidel Signless Laplacian matrix, Seidel energy, Seidel Signless Laplacian energy, Maximum and Minimum energy, Degree Sum energy and Distance Degree energy of the Unitary Cayley graphs [UCG] have been calculated. Low-power devices must be able to transfer data across long distances with low delay and reliability. To overcome this drawback a small-world network depending on the unitary Cayley graph is proposed to decrease the delay and increase the reliability and is also used to create and analyze network communication. Small-world networks based on the Cayley graph have a basic construction and are highly adaptable. The simulation result shows that the small-world network based on unitary Cayley graphs has a shorter delay and is more reliable. Furthermore, the maximum delay is lowered by 40%.

A note on the Seidel and Seidel Laplacian matrices

In this paper we investigate the spectrum of the Seidel and Seidel Laplacian matrix of a graph. We generalized the concept of Seidel Laplacian matrix which denoted by Seidel matrix and obtained some results related to them. This can be intuitively understood as a consequence of the relationship between the Seidel and Seidel Laplacian matrix in the graph by Zagreb index. In closing, we mention some alternatives to and generalization of the Seidel and Seidel Laplacian matrices. Also, we obtain relation between Seidel and Seidel Laplacian energy, related to all graphs with order n.

An improved lower bound for the Seidel energy of trees

Let G be a graph with the vertex set { v 1 , … , v n } . The Seidel matrix of G is an n × n matrix whose diagonal entries are zero, i j -th entry is − 1 if v i and v j are adjacent and otherwise is 1. The Seidel energy of G , denoted by E ( S ( G ) ) , is defined to be the sum of absolute values of all eigenvalues of the Seidel matrix of G . In Akbari et al. (2020), the authors proved that the Seidel energy of any graph of order n is at least 2 n − 2 . In this study, we improve the aforementioned lower bound for tree graphs.

On the Sα-matrix of graphs

Let [Formula: see text] and [Formula: see text] be the adjacency matrix and the diagonal matrix of vertex degrees of a graph [Formula: see text], respectively. For any real [Formula: see text], Nikiforov defined the matrix [Formula: see text] as [Formula: see text]. Let [Formula: see text] be complement of [Formula: see text]. Define [Formula: see text] where [Formula: see text] and [Formula: see text] are the identity matrix and the all-ones matrix, respectively. Since [Formula: see text] is the Seidel matrix of [Formula: see text], this new matrix is the generalization of Seidel matrix. Further, [Formula: see text] is a subset of universal adjacency matrices defined by Haemers and Omidi. In this paper, [Formula: see text]-eigenvalues and [Formula: see text]-energy of [Formula: see text] respectively, are investigated, which not only extends Seidel matrix but also enriches the theory of universal adjacency matrices.

The Seidel spectrum of two variants of join operations

The Seidel spectrum of a graph is defined as the multiset of all eigenvalues of its Seidel matrix. For two simple connected graphs [Formula: see text] and [Formula: see text], let us denote the subdivision-vertex join and subdivision-edge join by [Formula: see text] and [Formula: see text], respectively. In this paper, we completely determine the Seidel spectrum and corresponding Seidel eigenvectors of [Formula: see text] and [Formula: see text]. As an application, we give a sufficient and necessary condition for [Formula: see text] and [Formula: see text] to be Seidel integral.

Eigenvalue-free interval for Seidel matrices of threshold graphs

The distribution of eigenvalues for Seidel matrices of threshold graphs is investigated in this paper. We show that there is no eigenvalue of Seidel matrices of threshold graphs in the interval (−2,2) except for −1 and 1. We also determine the inertia of the Seidel matrix of a threshold graph in terms of its binary string.

Perfect codes and universal adjacency spectra of commuting graphs of finite groups

The commuting graph [Formula: see text] of a finite group [Formula: see text] has vertex set as [Formula: see text], and any two distinct vertices [Formula: see text] are adjacent if [Formula: see text] and [Formula: see text] commute with each other. In this paper, we first study the perfect codes of [Formula: see text]. We then find the universal adjacency spectra of the join of two regular graphs, join of two regular graphs in which one graph is a union of two regular graphs, and generalized join of regular graphs in terms of adjacency spectra of the constituent graphs and an auxiliary matrix. As a consequence, we obtain the adjacency, Laplacian, signless Laplacian, and Seidel spectra of the above graph operations. As an application of the results obtained, we calculate the adjacency, Laplacian, signless Laplacian, and Seidel spectra of [Formula: see text] for [Formula: see text], where [Formula: see text] is the dihedral group, [Formula: see text] is the dicyclic group and [Formula: see text] is the semidihedral group. Moreover, we provide the exact value of the spectral radius of the adjacency, Laplacian, signless Laplacian, and Seidel matrix of [Formula: see text] for [Formula: see text]. Some of the theorems published in [F. Ali and Y. Li, The connectivity and the spectral radius of commuting graphs on certain finite groups, Linear Multilinear Algebra 69 (2019) 1–14; T. Cheng, M. Dehmer, F. Emmert-Streib, Y. Li and W. Liu, Properties of commuting graphs over semidihedral groups, Symmetry 13(1) (2021) 103] can be deduced as corollaries from the theorems obtained in this paper.

On the spectra and eigenspaces of the universal adjacency matrices of arbitrary lifts of graphs

The universal adjacency matrix U of a graph Γ, with adjacency matrix A, is a linear combination of A, the diagonal matrix D of vertex degrees, the identity matrix I, and the all-1 matrix J with real coefficients, that is, , with and . Thus, in particular cases, U may be the adjacency matrix, the Laplacian, the signless Laplacian, and the Seidel matrix. In this paper, we develop a method for determining the universal spectra and bases of all the corresponding eigenspaces of arbitrary lifts of graphs (regular or not). As an example, the method is applied to give an efficient algorithm to determine the characteristic polynomial of the Laplacian matrix of the symmetric squares of odd cycles, together with closed formulas for some of their eigenvalues.

Some result on integrality of several matrix representation of complete r-uniform hypergraph

The eigenvalues of matrix representation hypergraph have the possibility that all of it can be an integer or otherwise. The Laplacian integral hypergraphs are those hypergraphs whose Laplacian spectrum consists entirely of integers likewise the definition of signless Laplacian integral and Seidel integral. This research focuses on the properties of the entry matrix and formulates the spectrum of the Laplacian matrix, Signless Laplacian matrix, and Seidel Matrix of complete r-uniform hypergraphs. Hence, it can determine the integrality of the representation matrix respectively. The result concludes that complete r-uniform hypergraphs are Laplacian integral, signless Laplacian integral, and Seidel integral.

On A-energy and S-energy of certain class of graphs

Abstract Let A and S be the adjacency and the Seidel matrix of a graph G respectively. A-energy is the ordinary energy E(G) of a graph G defined as the sum of the absolute values of eigenvalues of A. Analogously, S-energy is the Seidel energy ES(G) of a graph G defined to be the sum of the absolute values of eigenvalues of the Seidel matrix S. In this article, certain class of A-equienergetic and S-equienergetic graphs are presented. Also some linear relations on A-energies and S-energies are given.

Seidel Laplacian and Seidel Signless Laplacian Spectrum of the Zero-divisor Graph on the Ring of Integers Modulo <img src=image/13424080_01.gif>

Let <img src=image/13424080_02.gif> be a simple graph of order <img src=image/13424080_03.gif> and let <img src=image/13424080_04.gif> be the Seidel matrix of <img src=image/13424080_02.gif>, defined as <img src=image/13424080_05.gif> where <img src=image/13424080_06.gif> if the vertices <img src=image/13424080_07.gif> and <img src=image/13424080_08.gif> are adjacent and <img src=image/13424080_09.gif> if the vertices <img src=image/13424080_07.gif> and <img src=image/13424080_08.gif> are not adjacent and <img src=image/13424080_10.gif> if <img src=image/13424080_11.gif>. Let <img src=image/13424080_12.gif> be the diagonal matrix where <img src=image/13424080_13.gif> denotes the degree of the <img src=image/13424080_14.gif> vertex of <img src=image/13424080_02.gif>. The Seidel Laplacian matrix of a graph <img src=image/13424080_02.gif> is defined as <img src=image/13424080_15.gif> and the Seidel signless Laplacian matrix of a graph <img src=image/13424080_02.gif> is defined as <img src=image/13424080_16.gif>. The zero-divisor graph of a commutative ring <img src=image/13424080_17.gif>, denoted by <img src=image/13424080_18.gif>, is a simple undirected graph with all non-zero zero-divisors as vertices and two distinct vertices <img src=image/13424080_19.gif> are adjacent if and only if <img src=image/13424080_20.gif>. In this paper, we find the Seidel polynomial and Seidel Laplacian polynomial of the join of two regular graphs using the concept of schur complement and coronal of a square matrix. Also we describe the computation of the Seidel Laplacian and Seidel signless Laplacian eigenvalues of the join of more than two regular graphs, using the well known Fiedler's lemma and apply these results to describe these eigenvalues for the zero-divisor graph on <img src=image/13424080_21.gif>. Further we find the Seidel Laplacian and Seidel signless Laplacian spectrum of the zero-divisor graph of <img src=image/13424080_21.gif> for some values of <img src=image/13424080_03.gif>, say <img src=image/13424080_22.gif>, where <img src=image/13424080_23.gif> are distinct primes. We also prove that 0 is a simple Seidel Laplacian eigenvalue of <img src=image/13424080_24.gif>, for any <img src=image/13424080_03.gif>.

Maximality of Seidel matrices and switching roots of graphs

In this paper, we discuss maximality of Seidel matrices with a fixed largest eigenvalue. We present a classification of maximal Seidel matrices of largest eigenvalue 3, which gives a classification of maximal equiangular lines in a Euclidean space with angle \(\arccos 1/3\). Motivated by the maximality of the exceptional root system \(E_8\), we define strong maximality of a Seidel matrix, and show that every Seidel matrix achieving the absolute bound is strongly maximal.

Seidel energy for Cayley graphs associated to dihedral group

The studies on the energy of a graph have been actively studied since 1978 and there have been various types of energy of graphs proposed and studied by mathematicians all over the world. One of the many types of energy being studied is Seidel energy, where it is defined as the summation of the absolute values of the eigenvalues of the Seidel matrix of a graph. This research combines the study on three important branches in mathematics, i.e. energy of graph in linear algebra, Cayley graph in graph theory and dihedral group in group theory. The aim of this research is to present some values of Seidel energy of Cayley graphs associated to dihedral groups with respect to certain subsets of the group, namely the subsets of order one and the subsets of order two; {a, a n–1} for n ≥ 3 and {a 2, a n–2} for n ≥ 5. The results are proven by using some properties of special graphs such as complete graph, cycle graph and complete bipartite graph, including some relations between the eigenvalues of a graph, with the Seidel eigenvalues of a graph.