The signless Laplacian separator of a graph is defined as the difference between the largest eigenvalue and the second largest eigenvalue of the associated signless Laplacian matrix. In this paper, we determine the maximum signless Laplacian separators of unicyclic, bicyclic and tricyclic graphs with given order.

Abstract Symmetric matrices with zero row sums occur in many theoretical settings and real-life applications. When the offdiagonal elements of such are i.i.d. random variables large, eigenvalue distributions converge to a peculiar universal curve p_zrs(λ) that looks like cross between Wigner semicircle Gaussian distribution. An analytic theory for this curve, originally due Fyodorov, can be deve...

We give several bounds on the second smallest eigenvalue of the weighted Laplacian matrix of a finite graph and on the second largest eigenvalue of its weighted adjacency matrix. We establish relations between the given Cheeger-type bounds here and the known bounds in the literature. We show that one of our bounds is the best Cheeger-type bound available.

In this article we obtain optimal estimates for the eigenvalues of a natural operator KT# for locally strongly convex centro-affine and graph hypersurfaces. Several immediate applications of our eigenvalue estimates are presented. We also provide examples to illustrate that our eigenvalue estimates are optimal. MSC 2000: 53A15, 53B20, 53B25 (primary); 53C40 (secondary)

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; using a serial PC, it draws graphs of millions of nodes in less than a minute. ACE finds an optimal drawing by minimizing a quadratic energy function. The minimi...

Two results dealing with the relation between the smallest eigenvalue of a graph and its bipartite subgraphs are obtained. The first result is that the smallest eigenvalue μ of any non-bipartite graph on n vertices with diameter D and maximum degree ∆ satisfies μ > −∆ + 1 (D+1)n . This improves previous estimates and is tight up to a constant factor. The second result is the determination of th...

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. Moreover, the k largest eigenvalues of a random power law graph with exponent β have power l...

Let S be a certain set of graphs. A graph is called a minimizing graph in the set S if its least eigenvalue attains the minimum among all graphs in S . In this paper, we determine the unique minimizing graph in Gn, where Gn denotes the set of connected graphs of order n with cut vertices.