On the eigenvalues of a graphAllen SchwenkRobin J Wilson
Abstract It is shown that the second largest eigenvalue of the adjacency matrix of any d-regular graph G containing two edges the distance between which is at least 2k + 2 is at least 2d − 1 −(2d − 1 − 1)/(k+1).
The eigenvalues of multiplicity one are called the simple eigenvalues. The equation Ax=λx is called the eigenvalue equation of A, or of a labeled graph G if A=A(G). For a fixed λ∈Sp(G), its non-trivial solution x=(x1,x2,…,xn)T is a λ-eigenvector of A, or of a ...
As a consequence a random d-regular graph on n vertices, is, with high probability a certifiable efficient expander for n sufficiently large. The bound on the width of the interval is derived from martingale theory and the bound on E(λ2) is obtained by exploring the properties of random ...
Algebraic connectivity and the characteristic set of a graph Let Gbe a connected weighted graph on vertices {1,2,…,n} and L be the Laplacian matrix of GLet μ be the second smallest eigenvalue of L and Y be an eigen... R.,B.,Bapat,... - 《Linear & Multilinear Algebra》 被引量...
The graph HH with two vertices and a single edge between them has two eigenvalues: β1=−1β1=−1 and β2=1β2=1. If a graph GG has eigenvalues λ1≤λ2≤⋯≤λnλ1≤λ2≤⋯≤λn and contains the induced subgraph HH (that is, contains an edge), then the interlacing th...
On the distribution of eigenvalues of graphs Xuerong Yong Department of Mathematics, Xinjiang University, Urumqi 830046, P.R.China Abstract Let G be a simple graph with n(≥ 2) vertices, and λ i (G) be the ith largest eigenvalue of G. In this paper we obtain the following: If λ 3...
More spectral bounds on the clique and independence numbers We give some new bounds for the clique and independence numbers of a graph in terms of its eigenvalues. In particular we prove the following results. Let G... V Nikiforov - Academic Press, Inc. 被引量: 41发表: 2009年 On a Pos...
A Sharp Upper Bound for the Number of Spanning Trees of a Graph Let G=(V,E) be a simple graph with n vertices, e edges and d1 be the highest degree. Further let λi, i=1,2,...,n be the non-increasing eigenvalues of ... Kinkar,Ch.,Das - 《Graphs & Combinatorics》 被引量...
For a simple graph G with n-vertices, m edges and having Laplacian eigenvalues μ1,μ2,…,μn−1,μn=0, let Sk(G)=∑i=1kμi, be the sum of k largest Laplacian eigenvalues of G. Brouwer conjectured that Sk(G)≤m+(k+12), for all k=1,2,…,n. We obtain upper bounds for...