Eigenvalues Of Graph Research Articles

Bipartite unicyclic graphs with a unique perfect matching having the smallest positive eigenvalue equal to

The smallest positive eigenvalue τ ( G ) of a simple graph G is the smallest positive eigenvalue of its adjacency matrix A ( G ) . In [F. J. Zhang and A. Chang, Acyclic molecules with greatest HOMO-LUMO separation, Discrete Applied Mathematics, 98:165–171, (1999).], the authors characterized all nonsingular trees with τ equal to 2 − 1 . We consider the same problem for bipartite unicyclic graphs with a unique perfect matching. Let U be the class of all connected bipartite unicyclic graphs with a unique perfect matching. In this article, we characterize all graphs U in U with the property that τ ( U ) = 2 − 1 . Further, we show that the largest limit point of the smallest positive eigenvalues of graphs in U is 2 − 1 , whereas the smallest limit point is 0.

Complete equitable decompositions

A classical result in spectral graph theory states that if a graph G has an equitable partition π then the eigenvalues of the divisor graph Gπ are a subset of its eigenvalues, i.e. σ(Gπ)⊆σ(G). A natural question is whether it is possible to recover the remaining eigenvalues σ(G)−σ(Gπ) in a similar manner. Here we show that any weighted undirected graph with nontrivial equitable partition can be decomposed into a number of subgraphs whose collective spectra contain these remaining eigenvalues. Using this decomposition, which we refer to as a complete equitable decomposition, we introduce an algorithm for finding the eigenvalues of an undirected graph (symmetric matrix) with a nontrivial equitable partition. Under mild assumptions on this equitable partition we show that we can find eigenvalues of such a graph faster using this method when compared to standard methods. This is potentially useful as many real-world data sets are quite large and have a nontrivial equitable partition.

Exploration of regularities in bipartite graphs using GEOGEBRA software

A classroom proposal is presented to integrate contents of Graph Theory and Linear Algebra in complete bipartite graphs, linking adjacency and Laplacian matrices, the eigenvalues of graphs will be determined, applicable to connectivity concepts. Students will be given exploration activities working with GeoGebra software, starting from several particular cases, with table works and questionnaires to be completed, in order to determine patterns on the eigenvalues of adjacency and Laplacian matrices of complete bipartite graphs. The work with patterns will lead to the generalization process, to abstract properties from observation and experimentation on examples. This learning experience builds bridges between the concrete and the symbolic, and the student is initiated in research

Embedding and the first Laplace eigenvalue of a finite graph

Embedding and the first Laplace eigenvalue of a finite graph

The second smallest normalized Laplacian eigenvalue of unicyclic graphs

The second smallest normalized Laplacian eigenvalue of unicyclic graphs

Eigenvalues of complex unit gain graphs and gain regularity

Abstract A complex unit gain graph (or T {\mathbb{T}} -gain graph) Γ = ( G , γ ) \Gamma =\left(G,\gamma ) is a gain graph with gains in T {\mathbb{T}} , the multiplicative group of complex units. The T {\mathbb{T}} -outgain in Γ \Gamma of a vertex v ∈ G v\in G is the sum of the gains of all the arcs originating in v v . A T {\mathbb{T}} -gain graph is said to be an a a - T {\mathbb{T}} -regular graph if the T {\mathbb{T}} -outgain of each of its vertices is equal to a a . In this article, it is proved that a a - T {\mathbb{T}} -regular graphs exist for every a ∈ R a\in {\mathbb{R}} . This, in particular, means that every real number can be a T {\mathbb{T}} -gain graph eigenvalue. Moreover, denoted by Ω ( a ) \Omega \left(a) the class of connected T {\mathbb{T}} -gain graphs whose largest eigenvalue is the real number a a , it is shown that Ω ( a ) \Omega \left(a) is nonempty if and only if a a belongs to { 0 } ∪ [ 1 , + ∞ ) \left\{0\right\}\cup \left[1,+\infty ) . In order to achieve these results, non-complete extended p p -sums and suitably defined joins of T {\mathbb{T}} -gain graphs are considered.

Isospectral graphs via spectral bracketing

In this article, we develop a perturbative technique to construct families of non-isomorphic discrete graphs which are isospectral for the standard (also called normalised) Laplacian and its signless version. We use vertex contractions as a graph perturbation and spectral bracketing with auxiliary graphs which have certain eigenvalues with high multiplicity. There is no need to know explicitly the eigenfunctions of the corresponding graphs. In principle, one only needs to know the multiplicity of the eigenvalues of the auxiliary graphs, and that these eigenvalues are all different. We illustrate the method by presenting several families of examples of isospectral graphs including fuzzy balls, complete bipartite graphs and subdivision graphs obtained from the previous examples. All the examples constructed turn out to be also isospectral for the standard (Kirchhoff) Laplacian on the associated equilateral metric graph.

Distance Laplacian eigenvalues of graphs, and chromatic and independence number

Distance Laplacian eigenvalues of graphs, and chromatic and independence number

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 first two eigenvalues of regular graphs

Let G be a regular graph with m edges, and let μ1,μ2 denote the two largest eigenvalues of AG, the adjacency matrix of G. We show that, if G is not complete, thenμ12+μ22≤2(ω−1)ωm where ω is the clique number of G. This confirms a conjecture of Bollobás and Nikiforov for regular graphs. We also show that equality holds if and only if G is either a balanced Turán graph or the disjoint union of two balanced Turán graphs of the same size.

Mean-square output consensus for heterogeneous multi-agent systems over nonidentical packet loss channels

In this paper, we study the cooperative output consensus problem of discrete-time heterogeneous multi-agent systems over nonidentical channel fading networks. Different from the traditional output consensus problem, the communication channels among agents are perturbed by stochastic multiplicative noises, which can also describe the packet loss phenomenon. The perturbed channels can result in a significant challenge for consensus analysis since their effect cannot be separated linearly from the communication topology. Based on the distributed feedforward control method, a novel distributed dynamic output feedback controller is developed to ensure output synchronization for each agent in the mean-square sense. To facilitate the consensus analysis, we utilize the relationship between the graph Laplacian matrix and the incidence matrix and matrix transformation technique, and find sufficient solvability conditions relying on each agent’s dynamics, the maximum eigenvalue of communication graph and the packet loss probability for the scenarios of undirected graph and directed graph, respectively. Finally, simulation results are given to verify the theoretical analysis.

Some applications of eigenvalues of unitary Cayley graphs of matrix rings over finite fields

ABSTRACT In this work, we provide two applications of the eigenvalues of the unitary Cayley graphs over matrix rings over finite fields. For a ring R, denotes the Jacobson radical of R. In 2012, Kiani and Aghaei conjectured that if R and S are finite rings and their unitary Cayley graphs are isomorphic, then and are isomorphic. If this conjecture holds and , then R is characterized by its unitary Cayley graph, and we say that R is a ring determined by unitary Cayley graphs. Kiani and Aghaei showed that every finite commutative ring and are such rings. We examine the eigenvalues of and obtain many new families of rings determined by unitary Cayley graphs. In 2021, Podestá and Videla characterized all finite commutative rings R such that the graphs in the triple are mutually equienergetic non-isospectral and Ramanujan where is the unitary Cayley sum graph. We use the eigenvalues of the graph and some observation on the number of matrices of the given rank to extend Podestá and Videla's results by working on the finite non-commutative ring being a product of matrix rings over local rings.

An integrated exploration of heat kernel invariant feature and manifolding technique for 3D object recognition system

Spectral Graph theory has been utilized frequently in the field of Computer Vision and Pattern Recognition to address challenges in the field of Image Segmentation and Image Classification. In the proposed method, for classification techniques, the associated graph's Eigen values and Eigen vectors of the adjacency matrix or Laplacian matrix created from the images are employed. The Laplacian spectrum and a graph's heat kernel are inextricably linked. Exponentiating the Laplacian eigensystem over time yields the heat kernel, which is the solution to the heat equations. In the proposed technique K-Nearest neighborhood and Delaunay triangulation techniques are used to generate a graph from the 3D model. The graph is then represented into Normalized Laplacian (NL) and Laplacian matrix (L). From each Normalized Laplacian and Laplacian matrix, the feature vectors like Heat Content Invariant and Laplacian Eigen values are extracted. Then, using all of the available clustering algorithms on datasets, the optimum feature vector for clustering is determined. For clustering various manifolding techniques are employed. In the suggested method, the graph heat kernel is constructed using industry-standard objects which are taken from the Engineering bench mark Data set.

Some new results on the prime order Cayley graph of given groups

Abstract In this paper, we study the prime order Cayley graph assigned to the group ℤn for different values of n. We specify some of the graph theoretical properties such as chromatic and perfect matching numbers. Furthermore, we determine the adjacency matrices and eigenvalues of the prime order Cayley graph associated with groups ℤn and 𝒟2n.

Bordering of symmetric matrices and an application to the minimum number of distinct eigenvalues for the join of graphs

An important facet of the inverse eigenvalue problem for graphs is to determine the minimum number of distinct eigenvalues of a particular graph. We resolve this question for the join of a connected graph with a path. We then focus on bordering a matrix and attempt to control the change in the number of distinct eigenvalues induced by this operation. By applying bordering techniques to the join of graphs, we obtain numerous results on the nature of the minimum number of distinct eigenvalues as vertices are joined to a fixed graph.

The sum of the k largest distance eigenvalues of graphs

Motivated by progresses on study of eigenvalue sum in adjacency matrix and Laplacian matrix, in this paper, we focus on the sum of the k largest distance eigenvalues of graphs. Denote by Sk(D(G))=λ1(D(G))+⋯+λk(D(G)) the sum of the k largest distance eigenvalues of G. We initially determine the extremal graph attaining the minimum Sk(D(G)) among all threshold graphs except for complete graphs. Besides, we prove that the complete r-partite graph attains the minimum Sk(D(G)) among all graphs with chromatic number χ(G)=r and we also give the lower bounds on Sk(D(G)) when G is Kr+1-free, which are extensions of previous results from Lin (2019) and Lu & Lu (2022), respectively.

Proof of a conjecture on distribution of Laplacian eigenvalues and diameter, and beyond

Ahanjideh, Akbari, Fakharan and Trevisan proposed a conjecture on the distribution of the Laplacian eigenvalues of graphs: for any connected graph of order n with diameter d≥2 that is not a path, the number of Laplacian eigenvalues in the interval [n−d+2,n] is at most n−d. We show that the conjecture is true, and give a complete characterization of graphs for which the conjectured bound is attained. This establishes an interesting relation between the spectral and classical parameters.

On open neighborhood energy of graphs

The open neighborhood of a vertex u denoted by N(u) in a simple connected graph G is the collection of all vertices other than u and adjacent to u. In this paper, we introduce a square matrix of order n, called the open neighborhood matrix, ONM(G) of a graph G whose (i, j)th entry is |N(vi) ∩ N(vj)|/(di+dj) whenever vi~vj, i≠ j; and zero otherwise, where di and dj, are the degrees of vi and vj respectively. We then establish the relationship between the connectedness of the graph G and the multiplicity of the eigenvalue zero of the matrix ONM(G), if it exists. Furthermore, we found the bounds for the largest open neighbourhood eigenvalue and open neighbourhood energy of graphs.

Simple eigenvalues of cubic vertex-transitive graphs

Abstract If ${\mathbf v} \in {\mathbb R}^{V(X)}$ is an eigenvector for eigenvalue $\lambda $ of a graph X and $\alpha $ is an automorphism of X, then $\alpha ({\mathbf v})$ is also an eigenvector for $\lambda $ . Thus, it is rather exceptional for an eigenvalue of a vertex-transitive graph to have multiplicity one. We study cubic vertex-transitive graphs with a nontrivial simple eigenvalue, and discover remarkable connections to arc-transitivity, regular maps, and number theory.

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.