On the Grone-Merris conjecture
نویسنده
چکیده
One way to extract information about the structure of a graph is to encode the graph in a matrix and study the invariants of that matrix, such as the spectrum. In this note, we study the spectrum of the Combinatorial Laplacian matrix of a graph. The Combinatorial Laplacian of a simple graphG = (V,E) on the set of n vertices is the n×nmatrix L(G) that records the vertex degrees on its diagonal and−1 when an off-diagonal entry ij corresponds to an edge (i, j) ofG. The matrix L(G) is positive semidefinite, so its eigenvalues are real and non-negative. We list them in non-increasing order and with multiplicity: λ1(G) ≥ λ2(G) ≥ . . . ≥ λn(G) = 0 We are interested in the conjecture of Grone and Merris (GM) that the spectrum λ(L(G)) is majorized by the conjugate partition of the non-increasing sequence of vertex degrees of G (5). This question is currently being studied (see for example (4)), but has yet to be resolved. We extend the class of graphs for which the conjecture is known to hold to include trees. We also show that if GM holds for graph Laplacians, it also holds for more general Dirichlet Laplacians (cf. (2)) as conjectured by Duval (3).
منابع مشابه
Shifted Simplicial Complexes Are Laplacian Integral
We show that the combinatorial Laplace operators associated to the boundary maps in a shifted simplicial complex have all integer spectra. We give a simple combinatorial interpretation for the spectra in terms of vertex degree sequences, generalizing a theorem of Merris for graphs. We also conjecture a majorization inequality for the spectra of these Laplace operators in an arbitrary simplicial...
متن کاملNear Threshold Graphs
A conjecture of Grone and Merris states that for any graph G, its Laplacian spectrum, Λ(G), is majorized by its conjugate degree sequence, D∗(G). That conjecture prompts an investigation of the relationship between Λ(G) and D∗(G), and Merris has characterized the graphs G for which the multisets Λ(G) and D∗(G) are equal. In this paper, we provide a constructive characterization of the graphs G ...
متن کاملThe Grone-merris Conjecture
In spectral graph theory, the Grone-Merris Conjecture asserts that the spectrum of the Laplacian matrix of a finite graph is majorized by the conjugate degree sequence of this graph. We give a complete proof for this conjecture. The Laplacian of a simple graph G with n vertices is a positive semi-definite n×n matrix L(G) that mimics the geometric Laplacian of a Riemannian manifold; see §1 for d...
متن کاملA Majorization Bound for the Eigenvalues of Some Graph Laplacians
Grone and Merris [5] conjectured that the Laplacian spectrum of a graph is majorized by its conjugate vertex degree sequence. In this paper, we prove that this conjecture holds for a class of graphs including trees. We also show that this conjecture and its generalization to graphs with Dirichlet boundary conditions are equivalent.
متن کاملThe Grone Merris Conjecture and a Quadratic Eigenvalue Problem 3
Given a graph G = (V, E), we define the transpose degree sequence d T j to be equal to the number of vertices of degree at least j. We define L G , the graph Laplacian, to be the matrix, whose rows and columns are indexed by the vertex set V , whose diagonal entry at v is the degree of v and whose value at a pair (v, w) is −1 if (v, w) ∈ E and 0 otherwise. Grone and Merris conjectured [GM] Conj...
متن کامل