Eigenvalues Of Graph Research Articles

On normalized distance Laplacian eigenvalues of graphs and applications to graphs defined on groups and rings

The normalized distance Laplacian matrix of a connected graph $ G $, denoted by $ D^{\mathcal{L}}(G) $, is defined by $ D^{\mathcal{L}}(G)=Tr(G)^{-1/2}D^L(G)Tr(G)^{-1/2}, $ where $ D(G) $ is the distance matrix, the $D^{L}(G)$ is the distance Laplacian matrix and $ Tr(G)$ is the diagonal matrix of vertex transmissions of $ G. $ The set of all eigenvalues of $ D^{\mathcal{L}}(G) $ including their multiplicities is the normalized distance Laplacian spectrum or $ D^{\mathcal{L}} $-spectrum of $G$. In this paper, we find the $ D^{\mathcal{L}} $-spectrum of the joined union of regular graphs in terms of the adjacency spectrum and the spectrum of an auxiliary matrix. As applications, we determine the $ D^{\mathcal{L}} $-spectrum of the graphs associated with algebraic structures. In particular, we find the $ D^{\mathcal{L}} $-spectrum of the power graphs of groups, the $ D^{\mathcal{L}} $-spectrum of the commuting graphs of non-abelian groups and the $ D^{\mathcal{L}} $-spectrum of the zero-divisor graphs of commutative rings. Several open problems are given for further work.

Note on the sum of the smallest and largest eigenvalues of a triangle-free graph

Let G be a triangle-free graph on n vertices with adjacency matrix eigenvalues μ1(G)≥μ2(G)≥…≥μn(G). In this paper we study the quantityμ1(G)+μn(G). We prove that for any triangle-free graph G we haveμ1(G)+μn(G)≤(3−22)n. This was proved for regular graphs by Brandt, we show that the condition on regularity is not necessary. We also prove that among triangle-free strongly regular graphs the Higman-Sims graph achieves the maximum ofμ1(G)+μn(G)n.

Correction*

Abstract With this article in mind, we have found some results using eigenvalues of graph with sign. It is intriguing to note that these results help us to find the determinant of Normalized Laplacian matrix of signed graph and their coefficients of characteristic polynomial using the number of vertices. Also we found bounds for the lowest value of eigenvalue.

Eigenvalues of Cayley Graphs

We survey some of the known results on eigenvalues of Cayley graphs and their applications, together with related results on eigenvalues of Cayley digraphs and generalizations of Cayley graphs.

TEMO theorem for Sombor index

TEMO = topological effect on molecular orbitals was discovered by Polansky and Zander in 1982, in connection with the eigenvalues of molecular graphs. Eventually, analogous regularities were established for a variety of other topological indices. We now show that a TEMO-type regularity also holds for the Sombor index (\(SO\)): For the graphs \(S\) and \(T\), constructed by connecting a pair of vertex-disjoint graphs by two edges, \(SO(S) < SO(T)\) holds. Analogous relations are verified for several other degree-based graph invariants.

Critical parameters of the synchronisation's stability for coupled maps in regular graphs

Coupled Map Lattice (CML) models are particularly suitable to study spatially extended behaviours, such as wave-like patterns, spatio-temporal chaos, and synchronisation. Complete synchronisation in CMLs emerges when all maps have their state variables with equal magnitude, forming a spatially-uniform pattern that evolves in time. Here, we derive critical values for the parameters – coupling strength, maximum Lyapunov exponent, and link density – that control the synchronisation-manifold's linear stability of diffusively-coupled, identical, chaotic maps in generic regular graphs (i.e., graphs with uniform node degrees) and class-specific cyclic graphs (i.e., periodic lattices with cyclical node permutation symmetries). Our derivations are based on the Laplacian matrix eigenvalues, where we give closed-form expressions for the smallest non-zero eigenvalue and largest eigenvalue of regular graphs and show that these graphs can be classified into two sets according to a topological condition (derived from the stability analysis). We also make derivations for two classes of cyclic graph: k-cycles (i.e., regular lattices of even degree k, which can be embedded in Tk tori) and k-Möbius ladders, which we introduce here to generalise the Möbius ladder of degree k = 3. Our results highlight differences in the synchronisation manifold's stability of these graphs – even for identical node degrees – in the finite size and infinite size limit.

On normalized Laplacian eigenvalues of power graphs associated to finite cyclic groups

For a simple connected graph G of order n, the normalized Laplacian is a square matrix of order n, defined as [Formula: see text], where [Formula: see text] is the diagonal matrix whose i-th diagonal entry is [Formula: see text]. In this paper, we find the normalized Laplacian eigenvalues of the joined union of regular graphs in terms of the adjacency eigenvalues and the eigenvalues of quotient matrix associated with graph G. For a finite group [Formula: see text], the power graph [Formula: see text] of a group [Formula: see text] is defined as the simple graph in which two distinct vertices are joined by an edge if and only if one is the power of the other. As a consequence of the joined union of graphs, we investigate the normalized Laplacian eigenvalues of the power graphs of the finite cyclic group [Formula: see text].

On the minimum number of distinct eigenvalues of a threshold graph

For a graph G, we associate a family of real symmetric matrices, S(G), where for any A∈S(G), the location of the nonzero off-diagonal entries of A are governed by the adjacency structure of G. Let q(G) be the minimum number of distinct eigenvalues over all matrices in S(G). In this work, we give a characterization of all connected threshold graphs G with q(G)=2. Moreover, we study the values of q(G) for connected threshold graphs with trace 2, 3, n−2, n−3, where n is the order of threshold graph. The values of q(G) are determined for all connected threshold graphs with 7 and 8 vertices with two exceptions. Finally, a sharp upper bound for q(G) over all connected threshold graph G is given.

On the Eigenvalues of Grassmann Graphs, Bilinear Forms Graphs and Hermitian Forms Graphs

Motivated by conjectures of Karloff, and Van Dam and Sotirov on the smallest eigenvalues of Johnson and Hamming graphs, Brouwer, Cioabă, Ihringer and McGinnis obtained some new results involving the eigenvalues of various graphs coming from association schemes and posed some conjectures related to the eigenvalues of Grassmann graphs, Bilinear forms graphs and Hermitian forms graphs. In this paper, we prove some of their conjectures.

On the eccentricity spectra of threshold graphs

Threshold graphs have numerous applications in computer science and psychology. The eccentricity matrix ɛ(G) of G is defined as ɛ(G)ij=d(vi,vj)ifd(vi,vj)=min{e(vi),e(vj)},0otherwise.In this paper, we give some properties of the eccentricity eigenvalues of threshold graphs and determine threshold graphs with distinct eccentricity eigenvalues in [−2,0].

Conjugate Laplacian eigenvalues of co-neighbour graphs

Let G be a simple graph of order n. A vertex subset is called independent if its elements are pairwise non-adjacent. Two vertices in G are co-neighbour vertices if they share the same neighbours. Clearly, if S is a set of pairwise co-neighbour vertices of a graph G, then S is an independent set of G. Let c=a+b√m and c=a−b√m, where a and b are two nonzero integers and m is a positive integer such that m is not a perfect square. In [M. Lepovic, On conjugate adjacency matrices of a graph, Discrete Mathematics, 307, 730-738, 2007], the author defined the matrix Ac(G)=[cij]n to be the conjugate adjacency matrix of G, if cij=c for any two adjacent vertices i and j, cij=c for any two nonadjacent vertices i and j,and cij= 0 if i=j. In [S. Paul, Conjugate Laplacian matrices of a graph, Discrete Mathematics, Algorithms and Applications, 10, 1850082, 2018], the author defined the conjugate Laplacian matrix of graphs and described various properties of its eigenvalues and eigenspaces. In this article, we determine certain properties of the conjugate Laplacian eigenvalues and the eigenvectors of a graph with co-neighbour vertices.

Hamiltonian s-properties and eigenvalues of k-connected graphs

Chvátal and Erdös (1972) [5] proved that, for a k-connected graph G, if the stability number α(G)≤k−s, then G is Hamilton-connected (s=1) or Hamiltonian (s=0) or traceable (s=−1). Motivated by the result, we focus on tight sufficient spectral conditions for k-connected graphs to possess Hamiltonian s-properties. We say that a graph possesses Hamiltonian s-properties, which means that the graph is Hamilton-connected if s=1, Hamiltonian if s=0, and traceable if s=−1.For a real number a≥0, and for a k-connected graph G with order n, degree diagonal matrix D(G) and adjacency matrix A(G), we have identified best possible upper bounds for the spectral radius λ1(aD(Γ)+A(Γ)), where Γ is either G or the complement of G, to warrant that G possesses Hamiltonian s-properties. Sufficient conditions for a graph G to possess Hamiltonian s-properties in terms of upper bounds for the Laplacian spectral radius as well as lower bounds of the algebraic connectivity of G are also obtained. Other best possible spectral conditions for Hamiltonian s-properties are also discussed.

Group-Annihilator Graphs Realised by Finite Abelian Groups and Its Properties

Let G be a finite abelian group viewed a \({\mathbb {Z}}\)-module and let \({\mathcal {G}} = (V, E)\) be a simple graph. In this paper, we consider a graph \(\Gamma (G)\) called as a group-annihilator graph. The vertices of \(\Gamma (G)\) are all elements of G and two distinct vertices x and y are adjacent in \(\Gamma (G)\) if and only if \([x : G][y : G]G = \{0\}\), where \(x, y\in G\) and \([x : G] = \{r\in {\mathbb {Z}} : rG \subseteq {\mathbb {Z}}x\}\) is an ideal of a ring \({\mathbb {Z}}\). We discuss in detail the graph structure realised by a group G. Moreover, we study the creation sequence, hyperenergeticity and hypoenergeticity of group-annihilator graphs. Finally, we conclude the paper with a discussion on Laplacian eigen values of the group-annihilator graph. We show that the Laplacian eigen values are representatives of orbits of the group action: \(Aut(\Gamma (G)) \times G \rightarrow G\).

Spectral radii of friendship graphs and their connected induced subgraphs

The purpose of this work is to shed some new light on the spectral properties of the family of friendship graphs and, more generally, the family of weak-friendship graphs. A weak-friendship graph is a connected induced subgraph of a friendship graph. We characterize the spectral radius of a friendship graph and of each one of its connected induced subgraphs. We prove that the separability index of the family of weak-friendship graphs is equal to 2. We provide exact and asymptotic estimates for the number of complementarity eigenvalues of a weak-friendship graph.

On the smallest signless Laplacian eigenvalue of graphs

For a simple graph G the signless Laplacian matrix of G is defined as D(G)+A(G), where A(G) and D(G) are the adjacency matrix and the diagonal matrix of vertex degrees of G, respectively. By the smallest signless Laplacian eigenvalue of G, denoted by q′(G), we mean the smallest eigenvalue of the signless Laplacian matrix of G. In this paper we study the smallest signless Laplacian eigenvalue of graphs and find some relations between this and the chromatic number of graphs. We prove that if G is a graph of order n and with chromatic number χ(G), then q′(G)≤q′(T(n,χ(G))), where T(n,t) is the Turán graph on n vertices and t parts. Using this inequality we obtain some bounds for q′(G) that improve the known previous bounds.

Restructured class of estimators for population mean using an auxiliary variable under simple random sampling scheme

Abstract With this article in mind, we have found some results using eigenvalues of graph with sign. It is intriguing to note that these results help us to find the determinant of Normalized Laplacian matrix of signed graph and their coe cients of characteristic polynomial using the number of vertices. Also we found bounds for the lowest value of eigenvalue.

On the second largest eigenvalue of some Cayley graphs of the symmetric group

Let S_n and A_{n} denote the symmetric and alternating group on the set {1,ldots ,n}, respectively. In this paper we are interested in the second largest eigenvalue lambda _{2}(Gamma ) of the Cayley graph Gamma =mathrm{Cay}(G,H) over G=S_{n} or A_{n} for certain connecting sets H. Let 1<kle n and denote the set of all k-cycles in S_{n} by C(n, k). For H=C(n,n) we prove that lambda _{2}(Gamma )=(n-2)! (when n is even) and lambda _{2}(Gamma )=2(n-3)! (when n is odd). Further, for H=C(n,n-1) we have lambda _{2}(Gamma )=3(n-3)(n-5)! (when n is even) and lambda _{2}(Gamma )=2(n-2)(n-5) ! (when n is odd). The case H=C(n,3) has been considered in Huang and Huang (J Algebraic Combin 50:99–111, 2019). Let 1le r<k<n and let C(n,k;r) subseteq C(n,k) be set of all k-cycles in S_{n} which move all the points in the set {1,2,ldots ,r}. That is to say, g=(i_{1},i_{2},ldots ,i_{k})(i_{k+1})dots (i_{n})in C(n,k;r) if and only if {1,2,ldots ,r}subset {i_{1},i_{2},ldots ,i_{k}}. Our main result concerns lambda _{2}(Gamma ), where Gamma =mathrm{Cay}(G,H) with H=C(n,k;r) with 1le r<k<n when G=S_{n} if k is even and G=A_{n} if k is odd. Here we observe that λ2(Γ)≥(k-2)!n-rk-r1n-r((k-1)(n-k)-(k-r-1)(k-r)n-r-1).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\lambda _{2}(\\Gamma )\\ge (k-2)! {n-r \\atopwithdelims ()k-r} \\frac{1}{n-r} \\big ((k-1)(n-k) - \\frac{(k-r-1)(k-r)}{n-r-1}\\big ). \\end{aligned}$$\\end{document}We prove that this bound is attained in the special case k=r+1 , giving lambda _{2}(Gamma )=r!(n-r-1). The cases with H=C(n,3;1) and H=C(n,3;2) were considered earlier in [6].

Eigenvalues of Stochastic Blockmodel Graphs and Random Graphs with Low-Rank Edge Probability Matrices

We derive the limiting distribution for the outlier eigenvalues of the adjacency matrix for random graphs with independent edges whose edge probability matrices have low-rank structure. We show that when the number of vertices tends to infinity, the leading eigenvalues in magnitude are jointly multivariate Gaussian with bounded covariances. As a special case, this implies a limiting normal distribution for the outlier eigenvalues of stochastic blockmodel graphs and their degree-corrected or mixed-membership variants. Our result extends the classical result of Füredi and Komlós on the fluctuation of the largest eigenvalue for Erdős–Rényi graphs.

Adjacency eigenvalues of graphs without short odd cycles

It is well known that spectral Turán type problem is one of the most classical problems in graph theory. In this paper, we consider the spectral Turán type problem. Let G be a graph and let G be a set of graphs, we say G is G-free if G does not contain any element of G as a subgraph. Denote by λ1 and λ2 the largest and the second largest eigenvalues of the adjacency matrix A(G) of G, respectively. In this paper we focus on the characterization of graphs without short odd cycles according to the adjacency eigenvalues of the graphs. Firstly, an upper bound on λ12k+λ22k of n-vertex {C3,C5,…,C2k+1}-free graphs is established, where k is a positive integer. All the corresponding extremal graphs are identified. Secondly, a sufficient condition for non-bipartite graphs containing an odd cycle of length at most 2k+1 in terms of its spectral radius is given. At last, we characterize the unique graph having the maximum spectral radius among the set of n-vertex non-bipartite graphs with odd girth at least 2k+3, which solves an open problem proposed by Lin, Ning and Wu [Eigenvalues and triangles in graphs, Combin. Probab. Comput. 30 (2) (2021) 258-270].

Iterative learning control for consensus of second-order multi-agent systems with interval uncertain topologies

Abstract This paper is devoted to the robust consensus tracking problem of second-order nonlinear multi-agent systems (MASs) with the interval uncertain topologies. For the second-order MASs including one leader agent and multiple follower agents, a control protocol is proposed by combining the iterative learning control scheme with the sliding mode control method. By analyzing the convergence of sliding mode variables, the consensus conditions including the unknown eigenvalues and the undetermined weight coefficient are obtained. In order to deal with the difficulties of weight coefficient design caused by the unknown eigenvalues of graphs, a min-max optimization problem is formulated based on the fastest convergence of the λ-norm of sliding mode variables, then the optimal weight coefficient is obtained by solving the min-max optimization problem. Moreover, for the undirected and directed interval uncertain graphs, two algorithms about the optimal weight coefficients are proposed, respectively. Finally, three numerical simulation examples are presented to demonstrate the effectiveness of the proposed methods.