Partially answering a question of Paul Seymour, we obtain a sufficient eigenvalue condition for the existence of k edge-disjoint spanning trees in a regular graph, when k ∈ {2, 3}. More precisely, we show that if the second largest eigenvalue of a d-regular graph G is less than d − 2k−1 d+1 , then G contains at least k edge-disjoint spanning trees, when k ∈ {2, 3}. We construct examples of grap...

We present a construction of expander graphs obtained from Cayley graphs of narrow ray class groups, whose eigenvalue bounds follow from the Generalized Riemann Hypothesis. Our result implies that the Cayley graph of (Z/qZ)∗ with respect to small prime generators is an expander. As another application, we show that the graph of small prime degree isogenies between ordinary elliptic curves achie...

In this paper, we show that all fat Hoffman graphs with smallest eigenvalue at least −1−τ , where τ is the golden ratio, can be described by a finite set of fat (−1 − τ)-irreducible Hoffman graphs. In the terminology of Woo and Neumaier, we mean that every fat Hoffman graph with smallest eigenvalue at least −1−τ is anH-line graph, where H is the set of isomorphism classes of maximal fat (−1−τ)-...

Let U(n, k) be the set of non-bipartite unicyclic graphs with n vertices and k pendant vertices, where n ≥ 4. In this paper, the unique graph with the minimal least eigenvalue of the signless Laplacian among all graphs in U(n, k) is determined. Furthermore, it is proved that the minimal least eigenvalue of the signless Laplacian is an increasing function in k. Let Un denote the set of non-bipar...

Many graphs arising in various information networks exhibit the “power law” behavior – the number of vertices of degree k is proportional to k−β for some positive β. We show that if β > 2.5, the largest eigenvalue of a random power law graph is almost surely (1+o(1)) √ m where m is the maximum degree. When 2 < β < 2.5, the largest eigenvalue is heavily concentrated at cm3−β for some constant c ...

Known spectral methods for graph bipartition and image segmentation require numerical solution of eigenvalue problems with the graph Laplacian. We discuss several modern preconditioned eigenvalue solvers for computing the Fiedler vectors of large scale eigenvalue problems. The ultimate goal is to find a method with a linear complexity, i.e. a method with computational costs that scale linearly ...

Godsil showed that if Γ is a distance-regular graph with diameter D > 3 and valency k > 3, and θ is an eigenvalue of Γ with multiplicity m > 2, then k 6 (m+2)(m−1) 2 . In this paper we will give a refined statement of this result. We show that if Γ is a distance-regular graph with diameter D > 3, valency k > 2 and an eigenvalue θ with multiplicity m > 2, such that k is close to (m+2)(m−1) 2 , t...

A general method for obtaining asymptotic isoperimetric inequalities for families of graphs is developed. Some of these inequalities have been applied to functional analysis, This method uses the second smallest eigenvalue of a certain matrix associated with the graph and it is the discrete version of a method used before for Riemannian manifolds. Also some results are obtained on spectra of gr...

Alon and Roichman (1994) proved that for every ε > 0 there is a finite c(ε) such that for any sufficiently large group G, the expected value of the second largest (in absolute value) eigenvalue of the normalized adjacency matrix of the Cayley graph with respect to c(ε) log |G| random elements is less than ε. We reduce the number of elements to c(ε) logD(G) (for the same c), where D(G) is the su...

We present an extremely fast graph drawing algorithm for very large graphs, which we term ACE (for Algebraic multigrid Computation of Eigenvectors). ACE exhibits a vast improvement over the fastest algorithms we are currently aware of; it draws graphs of millions of nodes in less than a minute. ACE finds an optimal drawing by minimizing a quadratic energy function. The minimization problem is e...