In the last decade important relations between Laplace eigenvalues and eigenvectors of graphs and several other graph parameters were discovered. In these notes we present some of these results and discuss their consequences. Attention is given to the partition and the isoperimetric properties of graphs, the max-cut problem and its relation to semidefinite programming, rapid mixing of Markov chains, and to extensions of the results to infinite graphs.

. An independent set of three vertices such that each pair is joined by a path that avoids the neighborhood of the third is called an asteroidal triple. A graph is asteroidal triple-free (AT-free, for short) if it contains no asteroidal triples. The motivation for this investigation was provided, in part, by the fact that the asteroidal triple-free graphs provide a common generalization of interval, permutation, trapezoid, and cocomparability graphs. The main contribution of this work is to investigate and reveal fundamental structural properties of AT-free graphs. Specifically, we show that every connected AT-free graph contains a dominating pair, that is, a pair of vertices such that every path joining them is a dominating set in the graph. We then provide characterizations of AT-free graphs in terms of dominating pairs and minimal triangulations. Subsequently, we state and prove a decomposition theorem for AT-free graphs. An assortment of other properties of AT-free graphs is also p...

Given a graph G = (V, E) and a set of n pairs of vertices in V, we are interested in finding for each pair (ai, bi), a path connecting ai to bi, such that the set of n paths so found is edge-disjoint. (For arbitrary graphs the problem is Af7>-complete, although it is in T' if n is fixed.) We present a polynomial time randomized algorithm for finding the optimal number of edge disjoint paths (up to constant factors) in the random regular graph Gn,r, for r sufficiently large. (The graph is chosen first, then an adversary chooses the pairs of endpoints.) 1

It is well known that if a stochastic service system (such as a cellular network) is shared by users with different characteristics (such as differing handoff rates or call holding times), the overall system performance can be improved by denial of service requests even when the excess capacity exists. Such selective denial of service based on system state is defined as call admission. A recent paper suggested the use of Genetic Algorithms to find near-optimal call admission policies for cellular networks. In this paper, we define local call admission policies that make admission decisions based on partial state information. We search for the best local call admission policies for one-dimensional cellular networks using Genetic Algorithms and show that the performance of the best local policies is comparable to optima for small systems. We test our algorithm on larger systems and show that the local policies found outperform the maximum packing and best handoff reservation policies for...

Abstract not found

Given a graph G = (V, E) and a set of n pairs of vertices in V, we are interested in finding for each pair (ai, bi), a path connecting ai to bi, such that the set of n paths so found is edge-disjoint. (For arbitrary graphs the problem is AfP-complete, although it is in 7 > if n is fixed.) We present a polynomial time randomized algorithm for finding edge disjoint paths in an r-regular expander graph G. We show that if G has sufficiently strong expansion properties and r is sufficiently large then all sets of n = f(n/log n) pairs of vertices can be joined. This is within a constant factor of best possible.

The "Steiner minimal tree" (SMT) of a point set P is the shortest network of "wires" which will suffice to "electrically" interconnect P . The "minimum spanning tree" (MST) is the shortest such network when only intersite line segments are permitted. The "Steiner ratio" ae(P ) of a point set P is the length of its SMT divided by the length of its MST. It is of interest to understand which point set (or point sets) in R d have minimal Steiner ratio. In this paper, we introduce a point set in R d which we call the "d-dimensional sausage." The 1 and 2-dimensional sausages have minimal Steiner ratios 1 and p 3=2 respectively. (The 2-sausage is the vertex set of an infinite strip of abutting equilateral triangles. The 3sausage is an infinite number of points evenly spaced along a certain helix.) We present extensive heuristic evidence to support the conjecture that the 3-sausage also has minimal Steiner ratio (ß 0:784190373377122). Also: We prove that the regular tetrahedron minimize...

We introduce new classes of valid inequalities, called wheel inequalities, for the stable set polytope PG of a graph G. Each "wheel configuration" gives rise to two such inequalities. The simplest wheel configuration is an "odd" subdivision W of a wheel, and for these we give necessary and sufficient conditions for the wheel inequality to be facet-inducing for PW . Generalizations arise by allowing subdivision paths to intersect, and by replacing the "hub" of the wheel by a clique. The separation problem for these inequalities can be solved in polynomial time. 1 Introduction Let G = (V; E) be a simple connected graph with jV j = n 2 and jEj = m. A subset of V is called a stable set if it does not contain adjacent vertices of G. Let N be a stable set. The incidence vector of N is x 2 f0; 1g V such that x v = 1 if and only if v 2 N . The stable set polytope of G, denoted by PG , is the convex hull of incidence vectors of stable sets of G. Some well-known valid inequalities for PG ...

We present an optimal \Theta (n)-time algorithm for the selection of a subset of the vertices of an n-vertex plane graph G so that each of the faces of G is covered by (i.e. incident with) one or more of the selected vertices. At most bn=2c vertices are selected, matching the worst-case requirement. Analogous results for edge-covers are developed for two different notions of "coverage". In particular,our linear-time algorithm selects at most n \Gamma 2 edges to strongly cover G, at most bn=3c diagonals to cover G, and in the case where G has no quadrilateral faces, at most bn=3c edges to cover G. All these bounds are optimal in the worst-case. Most of our results flow from the study of a relaxation of thefamiliar notion of a 2-coloring of a plane graph which we call a face-respecting 2-coloring that permits

If π is a cyclic order of the vertices of a graph G, the number h(π) is defined to be the sum of the distances between consecutive vertices of G in π. For a graph G, the hamiltonian spectrum H(G) is the set of all numbers h(π). The hamiltonian number h(G) of G is the minimum number contained in H(G) and the upper hamiltonian number h + (G) is the maximum number contained in H(G). We determine hamiltonian spectra of cycles. We also show that the upper hamiltonian number of a graph G of order n and diameter d is at least n + ⌈d 2 /2 ⌉ − 1. The bound is tight for all pairs n and d. 1