A General Computational Approach for Counting Labeled Graphs

Estimating the size of the maximum matching is a canonical problem in graph analysis, and one that has attracted extensive study over a range of different computational models. We present improved streaming algorithms for approximating the size of maximum matching with sparse (bounded arboricity) graphs. 
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\n* (Insert-Only Streams) We present a one-pass algorithm that takes O(alpha log n) space and approximates the size of the maximum matching in graphs with arboricity alpha within a factor of O(alpha). This improves significantly upon the state-of-the-art tilde{O}(alpha n^{2/3})-space streaming algorithms, and is the first poly-logarithmic space algorithm for this problem. 
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\n* (Dynamic Streams) Given a dynamic graph stream (i.e., inserts and deletes) of edges of an underlying alpha-bounded arboricity graph, we present an one-pass algorithm that uses space tilde{O}(alpha^{10/3}n^{2/3}) and returns an O(alpha)-estimator for the size of the maximum matching on the condition that the number edge deletions in the stream is bounded by O(alpha n). For this class of inputs, our algorithm improves the state-of-the-art tilde{O}(\\alpha n^{4/5})-space algorithms, where the \\tilde{O}(.) notation hides logarithmic in n dependencies.
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\nIn contrast to prior work, our results take more advantage of the streaming access to the input and characterize the matching size based on the ordering of the edges in the stream in addition to the degree distributions and structural properties of the sparse graphs.

The regularity lemma of Szemeredi turned out to be the most powerful tool for studying the testability of graph properties in the dense graph model. In fact, as we argue in this paper, this lemma can be used in order to prove (essentially) all the previous results in this area. More precisely, a barrier for obtaining an efficient testing algorithm for a graph property P was having an efficient regularity lemma for graphs satisfying P. The problem is that for many natural graph properties (e.g. triangle freeness) it is known that a graph can satisfy P and still only have regular partitions of tower-type size. This means that there was no viable path for obtaining reasonable bounds on the query complexity of testing such properties. In this paper we consider the property of being induced C_4-free, which also suffers from the fact that a graph might satisfy this property but still have only regular partitions of tower-type size. By developing a new approach for this problem we manage to overcome this barrier and thus obtain a merely exponential bound for testing this property. This is the first substantial progress on a problem raised by Alon in 2001, and more recently by Alon, Conlon and Fox. We thus obtain the first example of an efficient testing algorithm that cannot be derived from an efficient version of the regularity lemma.

Enumeration of graphs with a heavy-tailed degree sequence

A graph property is any nonempty isomorphism-closed class of simple (finite or infinite) graphs. A graph property P is of finite character if a graph G has a property P if and only if every finite induced subgraph of G has a property P . Let P1,P2, . . . ,Pn be graph properties of finite character, a graph G is said to be (uniquely) (P1,P2, . . . ,Pn)partitionable if there is an (exactly one) partition {V1, V2, . . . , Vn} of V (G) such that G[Vi] ∈ Pi for i = 1, 2, . . . , n. Let us denote by R = P1◦P2◦ · · · ◦Pn the class of all (P1,P2, . . . ,Pn)-partitionable graphs. A property R = P1◦P2◦ · · · ◦Pn, n ≥ 2 is said to be reducible. We prove that any reducible additive graph property R of finite character has a uniquely (P1,P2, . . . ,Pn)-partitionable countable generating graph. We also prove that for a reducible additive hereditary graph property R of finite character there exists a weakly universal countable graph if and only if each property Pi has a weakly universal graph. 242 J. Bucko and P. Mihok

A note on extremal trees with degree conditions

Social networks generally display a positively skewed degree distribution and higher values for clustering coefficient and degree assortativity than would be expected from the degree sequence. For some types of simulation studies, these properties need to be varied in the artificial networks over which simulations are to be conducted. Various algorithms to generate networks have been described in the literature but their ability to control all three of these network properties is limited. We introduce a spatially constructed algorithm that generates networks with constrained but arbitrary degree distribution, clustering coefficient and assortativity. Both a general approach and specific implementation are presented. The specific implementation is validated and used to generate networks with a constrained but broad range of property values.

Large graphs are sometimes studied through their degree sequences (power law or regular graphs). We study graphs that are uniformly chosen with a given degree sequence. Under mild conditions, it is shown that sequences of such graphs have graph limits in the sense of Lov\'{a}sz and Szegedy with identifiable limits. This allows simple determination of other features such as the number of triangles. The argument proceeds by studying a natural exponential model having the degree sequence as a sufficient statistic. The maximum likelihood estimate (MLE) of the parameters is shown to be unique and consistent with high probability. Thus $n$ parameters can be consistently estimated based on a sample of size one. A fast, provably convergent, algorithm for the MLE is derived. These ingredients combine to prove the graph limit theorem. Along the way, a continuous version of the Erd\H{o}s--Gallai characterization of degree sequences is derived.

The greedy tree $\mathcal{G}(D)$ and the $\mathcal{M}$-tree $\mathcal{M}(D)$ are known to be extremal among trees with degree sequence $D$ with respect to various graph invariants. This paper provides a general theorem that covers a large family of invariants for which $\mathcal{G}(D)$ or $\mathcal{M}(D)$ is extremal. Many known results, for example on the Wiener index, the number of subtrees, the number of independent subsets and the number of matchings follow as corollaries, as do some new results on invariants such as the number of rooted spanning forests, the incidence energy and the solvability. We also extend our results on trees with fixed degree sequence $D$ to the set of trees whose degree sequence is majorised by a given sequence $D$, which also has a number of applications.

Greedy trees are constructed from a given degree sequence by a simple greedy algorithm that assigns the highest degree to the root, the second-, third-, ... highest degrees to the root's neighbors, and so on. They have been shown to maximize or minimize a number of different graph invariants among trees with a given degree sequence. In particular, the total number of subtrees of a tree is maximized by the greedy tree. In this work, we show that in fact a much stronger statement holds true: greedy trees maximize the number of subtrees of any given order. This parallels recent results on distance-based graph invariants. We obtain a number of corollaries from this fact and also prove analogous results for related invariants, most notably the number of antichains of given cardinality in a rooted tree.

Difference graphs

Minimal reducible bounds for induced-hereditary properties

Local properties of geometric graphs

Consider property testing on bounded degree graphs and let $\\varepsilon>0$\ndenote the proximity parameter. A remarkable theorem of Newman-Sohler (SICOMP\n2013) asserts that all properties of planar graphs (more generally hyperfinite)\nare testable with query complexity only depending on $\\varepsilon$. Recent\nadvances in testing minor-freeness have proven that all additive and monotone\nproperties of planar graphs can be tested in $poly(\\varepsilon^{-1})$ queries.\nSome properties falling outside this class, such as Hamiltonicity, also have a\nsimilar complexity for planar graphs. Motivated by these results, we ask: can\nall properties of planar graphs can be tested in $poly(\\varepsilon^{-1})$\nqueries? Is there a uniform query complexity upper bound for all planar\nproperties, and what is the "hardest" such property to test? We discover a\nsurprisingly clean and optimal answer. Any property of bounded degree planar\ngraphs can be tested in $\\exp(O(\\varepsilon^{-2}))$ queries. Moreover, there is\na matching lower bound, up to constant factors in the exponent. The natural\nproperty of testing isomorphism to a fixed graph needs\n$\\exp(\\Omega(\\varepsilon^{-2}))$ queries, thereby showing that (up to\npolynomial dependencies) isomorphism to an explicit fixed graph is the hardest\nproperty of planar graphs. The upper bound is a straightforward adapation of\nthe Newman-Sohler analysis that tracks dependencies on $\\varepsilon$ carefully.\nThe main technical contribution is the lower bound construction, which is\nachieved by a special family of planar graphs that are all mutually far from\neach other. We can also apply our techniques to get analogous results for\nbounded treewidth graphs. We prove that all properties of bounded treewidth\ngraphs can be tested in $\\exp(O(\\varepsilon^{-1}\\log \\varepsilon^{-1}))$\nqueries. Moreover, testing isomorphism to a fixed forest requires\n$\\exp(\\Omega(\\varepsilon^{-1}))$ queries.\n

We investigate to what extent the degree sequence of a directed network constrains the number of driver nodes. We develop a pair of algorithms that take a directed degree sequence as input and aim to output a network with the maximum or minimum number of driver nodes. We find an upper bound for the maximum and a lower bound for the minimum, and show that the algorithms achieve these bounds for all real and model networks, with few exceptions characterized by tiny system size and heterogeneous degree distributions. Applying these algorithms to a broad range of real networks, we show the gap between the upper and lower bounds can vary dramatically across different degree sequences. Thus, we introduce the notion of \bl{structural control} complexity to capture how much more difficult it is to control a specific network beyond what is required given its degree sequence, suggesting additional structure is present. Using model networks, we numerically and analytically investigate how typical features of the degree distribution affect the range of required driver nodes. We find that the minimum is determined by the number of sources or sinks, while the maximum is strongly affected by the presence of hubs.

We consider quasirandom properties for Cayley graphs of finite abelian groups. We show that having uniform edge-distribution (i.e., small discrepancy) and having large eigenvalue gap are equivalent properties for such Cayley graphs, even if they are sparse. This positively answers a question of Chung and Graham ["Sparse quasi-random graphs", Combinatorica 22 (2002), no. 2, 217-244] for the particular case of Cayley graphs of abelian groups, while in general the answer is negative.