Some recent papers of Fan Chung

Sparse quasirandom graphs
Combinatorica, to appear.
(with R. L. Graham)
Abstract:
Quasirandom graph properties form a large equivalence class of graph properties which
are all shared by random graphs.
In recent years, various aspects of these properties have been treated by
a number of authors.
Almost all of these results deal with dense
graphs, that is, graphs on n vertices having cn^2 edges for some
c > 0 as n approaches infinity.
In this paper, we extend our study of quasirandomness to sparse graphs,
i.e., graphs on n vertices with o(n^2) edges.
It will be shown that many of the quasirandom properties for dense graphs
have analogous for sparse graphs,
while others do not, at least not without additional hypotheses.
In general, sparse graphs are more difficult to deal with than dense graphs, due for ex
ample to the possible absence of certain local structures, such as 4cycles.

The average distances in random graphs with given expected degrees
(with L. Lu)
Abstract:
Random graph theory is used to
examine the "smallworld phenomenon" 
any two strangers are connected
through a short chain of mutual acquaintances.
We will show that
for certain families of
random graphs with given expected degrees, the average distance is
almost surely of order
log n / log \tilde d where
\tilde d is the weighted average of the sum of squares of the
expected degrees.
Of particular interest are
power law random graphs in which the number
of
vertices of degree k is proportional to 1/k^{β} for some fixed
exponent β . For the case of β > 3 , we prove that
the average distance of
the power law graphs is almost surely of order
log n / log \tilde d . However, many Internet, social and citation networks
are power law graphs with exponents in the range 2 < β < 3 for which the power
law random graphs
have
average distance almost surely of order log log n,
but have diameter of order log n
(provided having some mild constraints for the average distance and maximum degree).
In particular, these graphs
contain
a dense subgraph, that we call
the core, having n ^{c/\log log n }
vertices. Almost all vertices are within distance
log log n of the core although there are vertices at distance log n from the core.

Connected components in random graphs with given degree
sequences
(with L. Lu)
Abstract:
We consider a family of random graphs with a given
expected degree sequence.
Each edge is chosen independently with probability proportional to
the product of the expected degrees of its two endpoints.
We show that
the distribution of its connected components are
primarily determined by the average degree d, independent of
the weight distribution. We prove that the
size of the second largest component of such a random graph on
n vertices is at most (2+o(1)) log
n / log (1+(d1)^2) and this is best possible when
the expected average degree
d is large.

Guessing secrets
Electronic Journal of Combinatorics, 8,(2001), #R13.
(with Ronald Graham and F. Tom Leighton)
Abstract:
We consider a variant of the familiar ``20 questions" problem
in which one tries to discover the identity of some unknown ``secret" by asking
binary questions.
In this variation, there is now a set of two (or more) secrets. For each questio
n asked, an adversary gets to choose which of the secrets to use in supply
ing the
answer, which in any case must be truthful.
We will describe a number of algorithms for dealing with this problem, although
we are still far from a complete understanding of the situation. Problems of thi
s
type have recently arisen in connection with certain Internet traffic routing
applications.

Random evolution of massive graphs
Handbook on Massive Data Sets, (Eds. James Abello et al.), to appear.
Extended abstract appeared in FOCS 2001, 510519.
(with W. Aiello and L. Lu)
Abstract:
Many massive graphs (such as WWW graphs and Call graphs) share
certain universal characteristics which can be described by
socalled the ``power law".
In this paper, we will first
briefly survey the history and previous work on power law graphs.
Then we will give four
evolution models for generating power law graphs by adding
one node/edge at a time. We will show that
for any given edge density and desired distributions for indegrees
and outdegrees (not necessarily the same, but adhered to certain
general conditions), the resulting graph will almost surely satisfy
the power law and the in/outdegree conditions.
We will show that
our most general directed and undirected models
include nearly all known models as special cases.
In addition, we consider another crucial aspects of massive graphs
that is called ``scalefree" in the sense that the frequency of
sampling
(w.r.t. the growth rate)
is independent of the parameter of the resulting power
law graphs.
We will show that our evolution models generate
scalefree power law graphs.

On sparse sets hitting linear forms
Journal of Number Theory, to appear.
(with Paul Erdös and Ronald Graham)
Abstract:
Let a_1 < a_2 < . . . < a_s be given integers and let [n] denote
the set { 1,2, . . ., n } . In this note we investigate the size of
minimum subsets of [n] which intersect every set in [n]
of the form {a_1 x , a_2 x, . . ., a_s x } for some x in
[n] .
For the most part, we can only estimate this extremal density
although there is an interesting class of a_i 's for which we can find the
exact answer.

Combinatorics for the East model,
(with Persi Diaconis and Ronald Graham)
Abstract:
We study the number of configurations in the East model of
statistical physics. This may be pictured as sites in a line. The site at zero
is always occupied. The site at i > 0 can only be changed if site
i1 is occupied. If at most n occupied sites are permitted,
we establish upper and lower bounds of the form
2^{ n choose 2} n! c^n
where
c < 1 for the number of possible configurations.

A chip firing game and Dirichlet eigenvalues,
(with Robert Ellis)
Abstract:
We consider a variation of the chipfiring game in an induced subgraph S of
a graph G. Starting from a given chip configuration,
if a vertex $v$ has at least as many chips as its degree, we can fire v
by sending one chip along each edge from v to its neighbors. The game
continues until no vertex can be fired. We will give an upper bound,
in terms of Dirichlet eigenvalues, for
the number of firings needed before a game terminates. We also examine
the relations among three equinumerous families, the set of spanning
forests on S with roots in the boundary of S, a set of "critical"
configurations of chips, and a coset group, called the sandpile group
associated with S.

Distance realization problems with applications to Internet tomography
(with Mark Garrett, Ronald Graham and David Shallcross).
Abstract:
In recent years, a variety of graph optimization problems
have arisen in which the graphs involved are much too large for the usual
algorithms to be effective. In these cases, even though we
are not able to examine the entire graph (which may be
changing dynamically), we would still like
to deduce various properties of it, such as the size of
a connected component, the set of neighbors of a subset of vertices, etc.
In this paper, we study a class of problems which
arise in the study of Internet data traffic models.
Several graphtheoretic problems are formulated that relate to the
inference of network topology given the distances (as represented by
endtoend delay) between pairs of special nodes designated as network
monitors. Some properties of these problems (such as NPcompleteness)
are derived and illustrated. A number of heuristic algorithms are
described and some initial results from experimental implementations
of these heuristics are provided. These problems are of interest for
monitoring largescale networks, or supplementing network management
techniques that may fail due to the interaction, or lack of standard
cooperation, among the heterogeneous technologies comprising the network.

Discrete Green's functions,
J. Combinatorial Theory (A),
to appear,
(with S.T. Yau)
Abstract:
We study discrete Green's functions and their relationship with
discrete Laplace equations.
Several methods for deriving Green's functions are discussed.
Green's functions can be used to deal with
diffusiontype problems on graphs, such as
chipfiring, load balancing and discrete Markov chains.

The diameter of random sparse graphs,
Advances in Applied Math, to appear.
(with Linyuan Lu)
Abstract:
We consider the diameter of a random graph G(n,p)
for various ranges of p close to the phase transition
point for
connectivity. For a disconnected graph G, we use the convention
that the diameter of G is the maximum diameter of its
connected components.
We show that almost surely
the diameter of random graph G(n,p) is close to
log n / log (np) if np approaches infinity. Moreover if
np / log n = c > 8, then the diameter of G(n,p)
is concentrated on two values. In general, if
np / log n = c > c_0, the diameter is concentrated on at most
2\lfloor 1 / c_0 \rfloor + 4 values.
We also proved that the diameter
of G(n,p) is almost surely equal to
the diameter of its giant component if np > 3.6.

A random graph model for massive graphs,
Proceedings of the
Thirtysecond Annual ACM Symposium on Theory of Computing
(2000),
171180.
(with Bill Aiello and Linyuan Lu)
The complete paper version
has a different title
A random graph model for power law graphs,
Abstract:
We propose a random graph model
which is a special case of sparse random graphs
with given degree sequences which satisfy a power law.
This model involves only a small number of parameters,
called logsize and loglog growth rate.
These parameters capture some universal
characteristics of massive graphs.
Furthermore, from these parameters, various properties
of the graph can be derived.
For example, for certain ranges of the parameters,
we will compute the expected
distribution of the sizes of the connected components
which almost surely occur with high probability.
We will illustrate the consistency of our model with
the behavior of some massive graphs derived
from data in telecommunications.
We will also discuss the threshold function, the
giant component, and the evolution of random graphs in
this model.

On polynomials of spanning trees,
Annals of Combinatorics,
4 (2000), 1326.
(with C. Yang)
Abstract:
The Kirchhoff polynomial of a graph G is
the sum of weights of all spanning trees where the
weight of a tree is the product of all its edge weights,
considered as formal variables.
Kontsevich conjectured that when edge weights are assigned values in
a finite field F_q, for a prime power q,
the number of zeros of
the Kirchhoff polynomial of a graph G
is just a polynomial function of q.
Stanley verified this conjecture for some families of graphs and
further proposed several conjectures.
In this paper, we derive and prove explicit formulas for certain
graphs, thus confirming Stanley's conjectures.
We also consider several extensions and generalizations
of the conjecture of Kontsevich.

Dynamic location problems with limited lookahead,
Theoretical Computer Science, to appear.
(with Ron Graham)
Abstract:

Spanning trees in subgraphs of lattices,
Comtempory Math, to appear.
Abstract:
We examine the relations between
the heat kernel, the zeta function and the number of
spanning trees of a graph.
In particular, we show that
for an induced subgraph S in the 2dimensional lattice graph,
the number of spanning trees of S is bounded
above by a function in S and the size of the boundary of S.

An upper bound for the Turan number t_3(n,4),
Journal of Combinatorial Theory (A),
87 (1999), 381389.
(with Linyuan Lu)
Abstract:
Let t_r(n,r+1) denote the smallest integer m such that every runiform
hypergraph on n vertices with m+1 edges must
contain a complete graph on r+1 vertices. In this paper,
we improve the previous lower bound for t_3(n,4).

Coverings, heat kernels and spanning trees
Electronic Journal of Combinatorics,
6, (1999), #R12.
(with S.T. Yau).
Abstract:
We consider a graph G and a covering \tilde{G}
of G and we study
the relations of their eigenvalues and heat kernels.
We evaluate the heat kernel for an infinite $k$regular tree
and we examine the heat kernels
for general kregular graphs.
In particular, we show that a kregular graph on n vertices
has at most
(1+o(1)) \frac {2\log n}{kn \log k} \left( \frac{(k1)^{k1}} {(k^22k)^{k/21}}
\right)^n
spanning trees, which is
best possible within a constant factor.

Higher eigenvalues and isoperimetric inequalities on
Riemannian manifolds and graphs,
Communications on Analysis and Geometry,
to appear
(with A. Grigor'yan and S.T. Yau).

Augumented ring networks,
(with Bill Aiello, Sandeep Bhatt and Arny
Rosenberg, Ramesh Sitaraman)

Open problems of Paul Erdos in graph theory,
Journal of Graph Theory, 25 (1997), 336.

Multidiameters and multiplicities,
European Journal of Combinatorics,
20 (1999), 629640.
(with C. Delorme and P. Sol'e ).

Forced convex ngons in the plane,
Discrete and Computational Geometry,
19 (1998), 367371.
(with R.L. Graham)

Spectral Graph Theory, AMS Publications, 1997.
The first chapter is
here .

Stratified random walks on an $n$cube,
Random Structures and Algorithms,
11 (1997), 199224.
(with R.L. Graham)

Random walks on generating sets of groups,
Electronic Journal of Combinatorics,
4, no. 2, (1997) #R7
(with R. L. Graham).

Logarithmic Sobolev techniques for random walks on graphs .
Emerging Applications of Number Theory,
IMA Volumes in Math. and its Applications,
109 (eds. D. A. Hejhal et. al.), 175186, Springer, 1999.

Logarithmic Harnack inequalities,
Mathematical Research Letters, 3 (1996), 793812.
(with S.T. Yau)

Isoperimetric inequalities for cartesian products of graphs
,
Probability, Combinatorics and Computing,
7 (1998), 141148.
(with Prasad Tetali).

Weighted graph Laplacians and isoperimetric inequalities,
Pacific Journal of Mathematics,
192 (2000), 257274.
(with Kevin Oden)

On optimal strategies for stealing cycles,
IEEE Trans. on Computers, 46 (1997), 545557
(with S. Bhatt,
F. T. Leighton, and A. L. Rosenberg)

Maximum subsets of $(0,1]$ with no solutions to $x+y=kz$,
Electronic Journal of Combinatorics, 3 (1996) #1
(with John L. Goldwasser)

A Harnack inequality for Dirichlet eigenvalues,
Journal of Graph Theory,
34 (2000), 247257.
(with S.T. Yau).

A combinatorial trace formula,
Tsing Hua Lectures on Geometry and Analysis,
International Press, Cambridge, Massachusetts, 1997, 107116.
(with S.T. Yau)

A combinatorial Laplacian with vertex weights,
Journal of Combinatorial Theory, (A), 75 (1996) 316327.
(with R. P. Langlands)

Eigenvalues and diameters for manifolds and graphs,
Tsing Hua Lectures on Geometry and Analysis, .
International Press, Cambridge, Massachusetts, 1997, 79106.
(with A. Grigor'yan and S. T. Yau)

Optimal emulations by butterfly networks ,
JACM, 43 (1996), 316327.
(with S. Bhatt, JiaWei Hong,
F. T. Leighton, Bojana Obrenic, A. L. Rosenberg and E.J. schwabe)

Upper bounds for eigenvalues of the discrete and continuous
Laplace operators,
Advances in Mathematics,
117 (1996) 165178.
(with A. Grigor'yan and S. T. Yau)

On sampling with Markov chains,
Random Structures and Algorithms, 9 (1996) 5578.
(with R. L. Graham and S. T. Yau)

Eigenvalue inequalities for graphs and convex subgraphs,
Communications on Analysis and Geometry,
5 (1998), 575624.
(with S.T. Yau)

A Harnack inequality for homogeneous graphs and subgraphs,
Communications on Analysis and Geometry, 2
(1994), 628639
(with S.T. Yau)

Eigenvalues of graphs,
Proceedings of the International Congress of Mathematicians,
Zurich, (1994),
Birkh"auser Verlag, Berlin, 13331342.

Scheduling treedags using FIFO queues: A controlmemory tradeoff,
(with
Sandeep N. Bhatt, Tom Leighton and Arny Rosenberg)

Eigenvalues of graphs and Sobolev inequalities,
Combinatorics, Probability and Computing, . 4 (1995) 1126.
(with S. T. Yau)

Salvage embeddings of complete trees,
SIAM J. Discrete Math., 8 (1995), 617637
(with S. Bhatt, F. T. Leighton, and A. L. Rosenberg)

Groups and the Buckyball,
in Lie Theory and Geometry: In honor of Bertram Kostant
(Eds. J.L. Brylinski, R. Brylinski, V. Guillemin
and V. Kac)
PM 123, Birkh"auser, Boston, 1994
(with Bertram Kostant and Shlomo Sternberg)

On the cover polynomial of a digraph,
J. Combinatorial Theory, (B) 65 (1995), 273290
(with R. L. Graham)

Routing permutations on graphs via matchings,
SIAM J. Discrete Math. 7 (1994) 513530
(with Noga Alon and R. L. Graham)

Mathematics and the Buckyball,
(This is a somewhat different version from the one
that appeared in American Scientist, v. 81, No. 1., (1993)
5671)
(with Shlomo Sternberg)

Chordal completions of planar graphs,
J. of Comb. Th. (B) 62 (1994) 96106.
(with David Mumford)

A report on reliable software and communication,
Bellcore Internal Memorandom 1992.
Paper version titled `` Reliable software and communication: An overview",
IEEE J. Selected Areas Comm. v. 12, no. 1 (1994), 2332.

Laplacian and vibrational spectra for homogeneous graphs,
J. Graph Theory , 16 (1992), 605627
(with Shlomo Sternberg)

Communication complexity and quasirandomness,
SIAM J. Discrete Math. 6 (1993), 110123.
(with Prasad Tetali)

Several generalizations of Weil's sums,
J. of Number Theory, 49,
(1994) 95106.

Subgraphs of a hypercube containing no small cycles,
J. Graph Theory, 16 (1992), 273286

Even cycles in directed graphs,
SIAM J. Discrete Math. 7 (1994) 474483
(with Wayne Goddard and D. J. Kleitman)
Most of the rest of 140+ papers are not available electronically.
Reprints will be sent upon request.