Eigenvalue interlacing and weight parameters of graphs 1 (original) (raw)

C O ] 2 J un 2 01 2 Eigenvalue Interlacing and Weight Parameters of Graphs ∗

Eigenvalue interlacing is a versatile technique for deriving results in algebraic com-binatorics. In particular, it has been successfully used for proving a number of results about the relation between the (adjacency matrix or Laplacian) spectrum of a graph and some of its properties. For instance, some characterizations of regular partitions, and bounds for some parameters, such as the independence and chromatic numbers, the diameter, the bandwidth, etc., have been obtained. For each parameter of a graph involving the cardinality of some vertex sets, we can define its corresponding weight parameter by giving some " weights " (that is, the entries of the positive eigenvector) to the vertices and replacing cardinalities by square norms. The key point is that such weights " regularize " the graph, and hence allow us to define a kind of regular partition, called " pseudo-regular, " intended for general graphs. Here we show how to use interlac-ing for provi...

Eigenvalue interlacing and weight parameters of graphs

Linear Algebra and its Applications, 1999

Eigenvalue interlacing is a versatile technique for deriving results in algebraic combinatorics. In particular, it has been successfully used for proving a number of results about the relation between the (adjacency matrix or Laplacian) spectrum of a graph and some of its properties. For instance, some characterizations of regular partitions, and bounds for some parameters, such as the independence and chromatic numbers, the diameter, the bandwidth, etc., have been obtained. For each parameter of a graph involving the cardinality of some vertex sets, we can define its corresponding weight parameter by giving some "weights" (that is, the entries of the positive eigenvector) to the vertices and replacing cardinalities by square norms. The key point is that such weights "regularize" the graph, and hence allow us to define a kind of regular partition, called "pseudo-regular," intended for general graphs. Here we show how to use interlacing for proving results about some weight parameters and pseudo-regular partitions of a graph. For instance, generalizing a well-known result of Lovász, it is shown that the weight Shannon capacity Θ * of a connected graph Γ, with n vertices and (adjacency matrix) eigenvalues λ 1 > λ 2 ≥ • • • ≥ λ n , satisfies * Work supported in part by the Spanish Research Council (Comisión Interministerial de Ciencia y Tecnología, CICYT) under projects TIC 94-0592 and TIC 97-0963.

On the multiplicity of the eigenvalues of a graph

Acta Mathematica Hungarica, 2007

Given a graph G with characteristic polynomial ϕ(t), we consider the ML-decomposition ϕ(t) = q1(t)q2(t) 2 . . . qm(t) m , where each qi(t) is an integral polynomial and the roots of ϕ(t) with multiplicity j are exactly the roots of qj(t). We give an algorithm to construct the polynomials qi(t) and describe some relations of their coefficients with other combinatorial invariants of G. In particular, we get new bounds for the energy E(G) = n i=1 |λ i | of G, where λ 1 , λ 2 , . . . , λ n are the eigenvalues of G (with multiplicity). Most of the results are proved for the more general situation of a Hermitian matrix whose characteristic polynomial has integral coefficients. * This work was done during a visit of the second named author to UNAM.

A combinatorial bound on the number of distinct eigenvalues of a graph

2022

The smallest possible number of distinct eigenvalues of a graph G, denoted by q(G), has a combinatorial bound in terms of unique shortest paths in the graph. In particular, q(G) is bounded below by k, where k is the number of vertices of a unique shortest path joining any pair of vertices in G. Thus, if n is the number of vertices of G, then n − q(G) is bounded above by the size of the complement (with respect to the vertex set of G) of the vertex set of the longest unique shortest path joining any pair of vertices of G. The purpose of this paper is to commence the study of the minor-monotone floor of n − k, which is the minimum of n − k among all graphs of which G is a minor. Accordingly, we prove some results about this minor-monotone floor.

On the coefficient of λ in the characteristic polynomial of singular graphs

Utilitas Mathematica

A singular graph, with adjacency matrix A and one zero eigenvalue, has a corresponding eigenvector v 0 which is related to L, the coefficient of λ of the characteristic polynomial φ(G, λ) =Det(λI − A). In this paper a simple formula is derived expressing L in terms of the norm of v 0 . Furthermore it is shown that the ratio of the diagonal cofactors, which are the determinants of the adjacency matrices of the vertex-deleted subgraphs of G, can be obtained from a kernel eigenvector. The nonsingular vertex-deleted subgraphs of G are characterised. Results are also obtained for singular graphs with more than one zero eigenvalue.

On the multiplicity of Laplacian eigenvalues and Fiedler partitions

Linear Algebra and its Applications

In this paper we investigate the relation between eigenvalue distribution and graph structure of two classes of graphs: the (m, k)-stars and l-dependent graphs. We give conditions on the topology and edge weights in order to get values and multiplicities of Laplacian matrix eigenvalues. We prove that a vertex set reduction on graphs with (m, k)-star subgraphs is feasible, keeping the same eigenvalues with reduced multiplicity. Moreover, some useful eigenvectors properties are derived up to a product with a suitable matrix. Finally, we relate these results with Fiedler spectral partitioning of the graph and the physical relevance of the results is shortly discussed.

On main eigenvalues of certain graphs

2016

An eigenvalue of the adjacency matrix of a graph is said to be main if the all-1 vector is not orthogonal to the associated eigenspace. In this work, we approach the main eigenvalues of some graphs. The graphs with exactly two main eigenvalues are considered and a relation between those main eigenvalues is presented. The particular case of harmonic graphs is analyzed and they are characterized in terms of their main eigenvalues without any restriction on its combinatorial structure. We give a necessary and sufficient condition for a graph G to have −1 − λ min as an eigenvalue of its complement, where λ min denotes the least eigenvalue of G. Also, we prove that among connected bipartite graphs, K r,r is the unique graph for which the index of the complement is equal to −1 − λ min. Finally, we characterize all paths and all double stars (trees with diameter three) for which the smallest eigenvalue is non-main. Main eigenvalues of paths and double stars are identified.

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...

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.