On normalized Laplacian eigenvalues of power graphs associated to finite cyclic groups (original) (raw)
Related papers
2021
For a simple graph G, the generalized adjacency matrix Aα(G) is defined as Aα(G) = αD(G) + (1− α)A(G), α ∈ [0, 1], where A(G) is the adjacency matrix and D(G) is the diagonal matrix of vertex degrees of G. This matrix generalises the spectral theories of the adjacency matrix and the signless Laplacian matrix of G. In this paper, we find Aα-spectrum of the joined union of graphs in terms of spectrum of adjacency matrices of its components and the zeros of the characteristic polynomials of an auxiliary matrix determined by the joined union. We determine the Aα-spectrum of join of two regular graphs, the join of a regular graph with the union of two regular graphs of distinct degrees. As an applications, we investigate the Aα-spectrum of certain power graphs of finite groups.
Carpathian Journal of Mathematics, 2022
The normalized distance Laplacian matrix of a connected graph G, denoted by D L (G), is defined by D L (G) = T r(G) −1/2 D L (G)T r(G) −1/2 , where D(G) is the distance matrix, the D L (G) is the distance Laplacian matrix and T r(G) is the diagonal matrix of vertex transmissions of G. The set of all eigenvalues of D L (G) including their multiplicities is the normalized distance Laplacian spectrum or D L-spectrum of G. In this paper, we find the D 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 L-spectrum of the graphs associated with algebraic structures. In particular, we find the D L-spectrum of the power graphs of groups, the D L-spectrum of the commuting graphs of non-abelian groups and the D L-spectrum of the zero-divisor graphs of commutative rings. Several open problems are given for further work.
A note on sum of powers of the Laplacian eigenvalues of graphs
Applied Mathematics Letters, 2011
For a graph G and a real number α ̸ = 0, the graph invariant s α (G) is the sum of the αth power of the non-zero Laplacian eigenvalues of G. This note presents some bounds for s α (G) in terms of the vertex degrees of G, and a relation between s α (G) and the first general Zagreb index, which is a useful topological index and has important applications in chemistry.
On sum of powers of the Laplacian eigenvalues of graphs
Linear Algebra and its Applications, 2008
For a graph G and a real α / = 0, we study the graph invariant s α (G)-the sum of the αth power of the non-zero Laplacian eigenvalues of G. The cases α = 2, 1 2 and −1 have appeared in different problems. Here we establish some properties for s α with α / = 0, 1. We also discuss the cases α = 2, 1 2 .
Signless Laplacian spectrum of power graphs of finite cyclic groups
AKCE International Journal of Graphs and Combinatorics, 2019
In this paper, we have studied Signless Laplacian spectrum of the power graph of finite cyclic groups. We have showed that n − 2 is an eigen value of Signless Laplacian of the power graph of Z n , n ≥ 2 with multiplicity at least φ(n). In particular, using the theory of Equitable Partitions, we have completely determined the Signless Laplacian spectrum of power graph of Z n for n = pq where p, q are distinct primes. c
On the Laplacian eigenvalue 222 of graphs
2019
Let GGG be a graph. The Laplacian matrix of GGG is L(G)=D(G)−A(G)L(G)=D(G)-A(G)L(G)=D(G)−A(G), where D(G)=diag(d(v1),ldots,d(vn))D(G)=diag(d(v_{1}),\ldots , d(v_{n}))D(G)=diag(d(v1),ldots,d(vn)) is a diagonal matrix and d(v)d(v)d(v) denotes the degree of the vertex vvv in GGG and A(G)A(G)A(G) is the adjacency matrix of GGG. Let G1G_1G1 and G2G_2G2 be two (unicyclic) graphs. We study the multiplicity of the Laplacian eigenvalue 222 of G=G1odotG2G=G_1\odot G_2G=G1odotG2 where the graphs G1G_1G1 or G2G_2G2 may have perfect matching and Laplacian eigenvalue 222 or not. We initiate the Laplacian characteristic polynomial of G1G_1G1, G2G_2G2 and G=G1odotG2G=G_1\odot G_2G=G1odotG2. It is also investigated that Laplacian eigenvalue 222 of G=G1odotG2G=G_1\odot G_2G=G_1odotG2 for some graphs G1G_1G1 and G2G_2G_2 under the conditions.
The Least Eigenvalue of the Complement of the Square Power Graph of G
Complexity
Let G n , m represent the family of square power graphs of order n and size m , obtained from the family of graphs F n , k of order n and size k , with m ≥ k . In this paper, we discussed the least eigenvalue of graph G in the family G n , m c . All graphs considered here are undirected, simple, connected, and not a complete K n for positive integer n .
Spectrum and L-spectrum of the power graph and its main supergraph for certain finite groups
Filomat, 2017
Let G be a finite group. The power graph P(G) and its main supergraph S(G) are two simple graphs with the same vertex set G. Two elements x,y ? G are adjacent in the power graph if and only if one is a power of the other. They are joined in S(G) if and only if o(x)|o(y) or o(y)|o(x). The aim of this paper is to compute the characteristic polynomial of these graph for certain finite groups. As a consequence, the spectrum and Laplacian spectrum of these graphs for dihedral, semi-dihedral, cyclic and dicyclic groups were computed.
On Distribution of Laplacian Eigenvalues of Graphs
2021
The work in this thesis concerns the investigation of eigenvalues of the Laplacian matrix, normalized Laplacian matrix, signless Laplacian matrix and distance signless Laplacian matrix of graphs. In Chapter 1, we present a brief introduction of spectral graph theory with some definitions. Chapter 222 deals with the sum of $ k $ largest Laplacian eigenvalues $ S_{k}(G) $ of graph $ G $ and Brouwer's conjecture. We obtain the upper bounds for $ S_{k}(G) $ for some classes of graphs and use them to verify Brouwer's conjecture for these classes of graphs. Also, we prove Brouwer's conjecture for more general classes of graphs. In Chapter 333, we investigate the Laplacian eigenvalues of graphs and the Laplacian energy conjecture for trees. We prove the Laplacian energy conjecture completely for trees of diameter $ 4 .Further,weprovethisconjectureforalltreeshavingatmost. Further, we prove this conjecture for all trees having at most .Further,weprovethisconjectureforalltreeshavingatmost \frac{9n}{25}-2 $ non-pendent vertices. Also, we obtain the sufficient conditions for the truth...