Jonathan Cutler

The Interlace Polynomial of Graphs at - 1

In this paper we give an explicit formula for the interlace polynomial atx=?1 for any graph, and as a result prove a conjecture of Arratia et al. that states that it is always of the form ± 2s. We also give a description of the graphs for which s is maximal.

Extremal graphs for homomorphisms

The study of graph homomorphisms has a long and distinguished history, with applications in many areas of graph theory. There has been recent interest in counting homomorphisms, and in particular on the question of finding upper bounds for the number of homomorphisms from a graph G into a fixed image graph H. We introduce our techniques by proving that the lex graph has the largest number of homomorphisms into K2 with one looped vertex (or equivalently, the largest number of independent sets) among graphs with fixed number of vertices and edges. Our main result is the solution to the extremal problem for the number of homomorphisms into P**math-image**, the completely looped path of length 2 (known as the Widom–Rowlinson model in statistical physics). We show that there are extremal graphs that are threshold, give explicitly a list of five threshold graphs from which any threshold extremal graph must come, and show that each of these “potentially extremal” threshold graphs is in fact extremal for some number of edges. Copyright

Negative dependence and srinivasan's sampling process

Dubhashi, Jonasson and Ranjan Dubhashi, Jonasson and Ranjan (2007) study the negative dependence properties of Srinivasans sampling processes (SSPs), random processes which sample sets of a fixed size with prescribed marginals. In particular they prove that linear SSPs have conditional negative association, by using the Feder–Mihail theorem and a coupling argument. We consider a broader class of SSPs that we call tournament SSPs (TSSPs). These have a tree-like structure and we prove that they have conditional negative association. Our approach is completely different from that of Dubhashi, Jonasson and Ranjan. We give an abstract characterization of TSSPs, and use this to deduce that certain conditioned TSSPs are themselves TSSPs. We show that TSSPs have negative association, and hence conditional negative association. We also give an example of an SSP that does not have negative association.

Hypergraph independent sets

The study of extremal problems related to independent sets in hypergraphs is a problem that has generated much interest. There are a variety of types of independent sets in hypergraphs depending on the number of vertices from an independent set allowed in an edge. We say that a subset of vertices is j-independent if its intersection with any edge has size strictly less than j. The Kruskal-Katona theorem implies that in an r-uniform hypergraph with a fixed size and order, the hypergraph with the most r-independent sets is the lexicographic hypergraph. In this paper, we use a hypergraph regularity lemma, along with a technique developed by Loh, Pikhurko and Sudakov, to give an asymptotically best possible upper bound on the number of j-independent sets in an r-uniform hypergraph.

Minimizing the number of independent sets in triangle-free regular graphs

Abstract Recently, Davies, Jenssen, Perkins, and Roberts gave a very nice proof of the result (due, in various parts, to Kahn, Galvin–Tetali, and Zhao) that the independence polynomial of a d -regular graph is maximized by disjoint copies of K d , d . Their proof uses linear programming bounds on the distribution of a cleverly chosen random variable. In this paper, we use this method to give lower bounds on the independence polynomial of regular graphs. We also give a new bound on the number of independent sets in triangle-free cubic graphs.

Extremal Graphs for Homomorphisms II: EXTREMAL GRAPHS FOR HOMOMORPHISMS II

Extremal problems for graph homomorphisms have recently become a topic of much research. Let hom(G,H) denote the number of homomorphisms from G to H. A natural set of problems arises when we fix an image graph H and determine which graph(s) G on n vertices and m edges maximize hom(G,H). We prove that if H is loop-threshold, then, for every n and m, there is a threshold graph G with n vertices and m edges which maximizes hom(G,H). Similarly, we show that loop-quasi-threshold image graphs have quasi-threshold extremal graphs. In the case H = P o 3 , the path on three vertices in which every vertex in looped, the authors [5] determined a set of five graphs, one of which must be extremal for hom(G,P o 3 ). Also in this paper, using similar techniques, we determine a set of extremal graphs for “the fox”, a graph formed by deleting the loop on one of the end-vertices of P o 3 . The fox is the unique connected loop-threshold image graph on at most three vertices for which the extremal problem was not previously solved.

The Maximum Number of Complete Subgraphs of Fixed Size in a Graph with Given Maximum Degree

In this paper, we make progress on a question related to one of Galvin that has attracted substantial attention recently. The question is that of determining among all graphs

The Multistep Friendship Paradox

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On the interlace polynomials of forests

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Counting dominating sets and related structures in graphs

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