## 779 Citations

### Connectedness in graph limits

- Mathematics
- 2008

We define direct sums and a corresponding notion of connectedness for graph limits. Every graph limit has a unique decomposition as a direct sum of connected components. As is well-known, graph…

### Interval Graph Limits

- MathematicsAnnals of combinatorics
- 2013

A graph limit theory for dense interval graphs is work out, which departs from the usual description of a graph limit as a symmetric function W(x, y) on the unit square, with x and y uniform on the interval.

### Interval Graph Limits

- MathematicsAnnals of Combinatorics
- 2012

We work out a graph limit theory for dense interval graphs. The theory developed departs from the usual description of a graph limit as a symmetric function W(x, y) on the unit square, with x and y…

### Independence and chromatic densities of graphs

- Mathematics
- 2011

We consider graph densities in countably inflnite graphs. The independence density of a flnite graph G of order n is its proportion of independent sets to all subsets of vertices, while the chromatic…

### Vertex Order in Some Large Constrained Random Graphs

- MathematicsSIAM J. Math. Anal.
- 2016

This paper proves that, under an assumption that holds for a wide range of parameter values, the constrained maximizers are in some sense monotone.

### Percolation on dense graph sequences.

- Mathematics
- 2010

In this paper we determine the percolation threshold for an arbitrary sequence of dense graphs (G n ). Let λn be the largest eigenvalue of the adjacency matrix of G n , and let G n (p n ) be the…

### Graph limits and parameter testing

- MathematicsSTOC '06
- 2006

We define a distance of two graphs that reflects the closeness of both local and global properties. We also define convergence of a sequence of graphs, and show that a graph sequence is convergent if…

### Monotone Graph Limits and Quasimonotone Graphs

- MathematicsInternet Math.
- 2012

It is shown that the kernel is monotone (i.e., increasing in both variables) if and only if the sequence satisfies a “quasimonotonicity” property defined by a certain functional tending to zero.

### Connectivity of inhomogeneous random graphs

- Mathematics, Computer ScienceRandom Struct. Algorithms
- 2014

The results generalize the classical result for G(n, p), when p=clogn/n and show that, under reasonably weak assumptions, the connectivity threshold of the model can be determined.

## References

SHOWING 1-10 OF 14 REFERENCES

### Strong independence of graph copy functions

- Mathematics
- 1978

Let H be a finite graph on v vertices . We define a function CH , with domain the set of all finite graphs, by letting cH(G) denote the fraction of subgraphs of G on v vertices isomorphic to H. Our…

### Strong Independence of Graphcopy Functions

- Mathematics
- 1979

Let H be a finite graph on v vertices . We define a function CH , with domain the set of all finite graphs, by letting cH(G) denote the fraction of subgraphs of G on v vertices isomorphic to H. Our…

### Recurrence of Distributional Limits of Finite Planar Graphs

- Mathematics
- 2000

Suppose that $G_j$ is a sequence of finite connected planar graphs, and in each $G_j$ a special vertex, called the root, is chosen randomly-uniformly. We introduce the notion of a distributional…

### Asymptotic Enumeration of Spanning Trees

- Mathematics, Computer ScienceCombinatorics, Probability and Computing
- 2005

It is shown that tree entropy is a logarithm of a normalized determinant of the graph Laplacian for infinite graphs, which is also expressed using random walks.

### Reflection positivity, rank connectivity, and homomorphism of graphs

- Mathematics
- 2004

It is shown that a graph parameter can be realized as the number of homomorphisms into a fixed (weighted) graph if and only if it satisfies two linear algebraic conditions: reflection positivity and…

### Quick Approximation to Matrices and Applications

- Computer ScienceComb.
- 1999

The matrix approximation is generalized to multi-dimensional arrays and from that derive approximation algorithms for all dense Max-SNP problems and the Regularity Lemma is derived.

### Quasi-random graphs

- Mathematics, Computer ScienceComb.
- 1989

A large equivalence class of graph properties is introduced, all of which are shared by so-called random graphs, and it is often relatively easy to verify that a particular family of graphs possesses some property in this class.

### Lower bounds of tower type for Szemerédi's uniformity lemma

- Mathematics, Computer Science
- 1997

This paper shows that the bound is necessarily of tower type, obtaining a lower bound given by a tower of 2s of height proportional to $ \log{(1/ \epsilon)} $).

### TRICKS OR TREATS WITH THE HILBERT MATRIX

- Mathematics
- 1983

It is natural to ask: What are the natural consequences from such a natural assemblage of (the reciprocals of) natural numbers? Hence, here arise ten concrete problems (instead of theorems), aimed to…