The Language of Self-Avoiding Walks

We study self-avoiding walk on graphs whose automorphism group has a transitive nonunimodular subgroup. We prove that self-avoiding walk is ballistic, that the bubble diagram converges at criticality, and that the critical two-point function decays exponentially in the distance from the origin. This implies that the critical exponent governing the susceptibility takes its mean-field value, and hence that the number of self-avoiding walks of length $n$ is comparable to the $n$th power of the connective constant. We also prove that the same results hold for a large class of repulsive walk models with a self-intersection based interaction, including the weakly self-avoiding walk. All of these results apply in particular to the product $T_{k}\times\mathbb{Z}^{d}$ of a $k$-regular tree ($k\geq3$) with $\mathbb{Z}^{d}$, for which these results were previously only known for large $k$.

By a nilpotent map we mean an orientably regular map whose orientation preserving automorphism group is nilpotent. The nilpotent maps are concluded to the maps whose automorphism group is a $2$-group and a complete classification of nilpotent maps of (nilpotency) class $2$ is given by Malnič et al. in [European J. Combin. 33 (2012), 1974-1986]. It is proved by Conder et al. in [J. Algebraic Combin. 44 (2016), 863-874] that given the class, there are finitely many simple nilpotent maps. However, for the nilpotent maps with multiple edges and given class, since its automorphism group may be infinitely big, it is impossible to list it by a computer. Therefore, to classify the nilpotent maps with small class $c$ is necessary and interesting. In this paper, the nilpotent maps of class $4$ will be determined.

Graph neural networks (GNNs) have revolutionized deep learning on non-Euclidean data domains, and are extensively used in fields such as social media and recommendation systems. However, complex relational data structures such as hypergraphs, pose challenges for GNNs in terms of their ability to model, embed, and learn relational complexities of multigraphs. Most GNNs focus on capturing flat local neighborhoods of a node thus failing to account for structural properties of multi-relational graphs. This paper introduces Hypergraph Link Prediction (HLP), a novel approach of encoding the multilink structure of graphs. HLP allows pooling operations to incorporate a 360 degrees overview of a node interaction profile, by learning local neighborhood and global hypergraph structure simultaneously. Global graph information is injected into node representations, such that unique global structural patterns of every node are encoded at the node level. HLP leverages the augmented hypergraph adjacency matrix to incorporate the depth of the hypergraph in the convolutional layers. The model is applied to the task of predicting multi-drug interactions, by modeling relations between pairs of drugs as a hypergraph. The existence and the type of drug interactions between the same pair of drugs are mapped as multiple edges, and can be inferred by learning the multigraph local and global structure concurrently. To account for molecular graph properties of a drug, additional drug chemical graph structural fingerprints are included as node attributes.

In this note we discuss extensions of results [5], where transient random walks are considered, whose transition matrix is compatible with a tree-structure of the underlying discrete state space. Notation is generally as in [5], with the exception that instead of a tree T we consider an arbitrary graph Γ, which is locally finite, connected, and whose vertex set S is infinite. The edge set E is unoriented, there are no multiple edges. We consider a transition operator P, given by a stochastic matrix (p(u, v))u, v ∈ S, which gives rise to a transient Markov chain (“random walk”), related to the graph structure by:(i) if p(u, v)>0 then [u, v] ∈ E,(ii) there is an M>0 such that p(k)(u, v)>0 for some k=k(u, v)≦M whenever [u, v]∈E (“uniform irreducibility”).

Following this advice, we should look over the big questions, then doodle and sketch out some approaches. If you are an expert, then this is easy to do, but most people do not want to wait to become an expert before looking at interesting problems. Graph theory, our area of expertise, has many hard-to-solve questions. Some hark back to the recreational roots of the area yet still keep their mystery. These ‘‘acorns’’ can be planted on the backs of envelopes, on a blackboard, and over a coffee. Our goal is to collect some of these conjectures—arguably some of the most intriguing—in one place. We present ten conjectures in graph theory, and you can read about each one in at most ten minutes. As we live in the era of Twitter, all the conjectures we state are 140 characters or fewer (so ‘‘minute’’ here has a double meaning). We might even call these sketchy tweets, as we present examples for each conjecture that you can doodle on as you read. Hamming also references ambiguity: good researchers can work both on proving and disproving the same statement, so we approach the conjectures with an open mind. He also mentions that a good approach is to reframe the problem, and change the point of view. One example from Vizing’s Conjecture (which is discussed as our second-to-last conjecture in the following text), is the three-page paper [2] which, with a new way of thinking, reduced most of the published work of twenty years to a corollary of its main result! Given the size of modern graph theory, with its many smaller subfields (such as structural graph theory, random graphs, topological graph theory, graph algorithms, spectral graph theory, graph minors, and graph homomorphisms, to name a few), it would be impossible to list all, or even the bulk of the conjectures in the field. We are content instead to focus on a few family jewels, which have an intrinsic beauty and have provided some challenges for graph-theorists for at least two decades. There is something for everyone here, from undergraduate students taking their first course in graph theory, to seasoned researchers in the field. For additional reading on problems and conjectures in graph theory and other fields, see the Open Problem Garden maintained by IRMACS at Simon Fraser University [24]. We consider only finite and undirected graphs, with no multiple edges or loops (unless otherwise stated). We assume the reader has some basic familiarity with graphs and their terminology, including notions such as cycles, paths, complete graphs, complete bipartite graphs, vertex degrees, and connected graphs. We use the notation Cn for the cycle with n vertices, Pn for the path with n vertices, and Kn for the complete graph with n vertices. The complete bipartite graph with m and n vertices of the respective colors is denoted by Km,n. All the background we need can be found in any text in graph theory, such as those of Diestel [9] and West [41], or online (see for example [42]).

Our word posets have finite words of bounded length as their elements, with the words composed from a finite alphabet. Their partial ordering follows from the inclusion of a word as a subsequence of another word. The elemental combinatorial properties of such posets are established. Their automorphism groups are determined (along with similar result for the word poset studied by Burosch, Frank and Röhl [4]) and a BLYM inequality is verified (via the normalized matching property).

We consider the problem of when a symbolic dynamical system supports a Borel probability measure that is invariant under every element of its automorphism group. It follows readily from a classical result of Parry that the full shift on finitely many symbols, and more generally any mixing subshift of finite type, supports such a measure. Frisch and Tamuz recently dubbed such measures characteristic, and further showed that every zero entropy subshift has a characteristic measure. While it remains open if every subshift over a finite alphabet has a characteristic measure, we define a new class of shifts, which we call language stable subshifts, and show that these shifts have characteristic measures. This is a large class that is generic in several senses and contains numerous positive entropy examples.

Spring algorithms are effective tools for visualizing undirected graphs. One major feature of applying spring algorithms is to display symmetric structures of graphs. This feature has been confirmed by numerous experiments. In this paper, we firstly formalize the concepts of graph symmetries in terms of “reflectional” and “rotational” automorphisms; and characterize the types of symmetries, which can be displayed simultaneously by a graph layout, in terms of “geometric” automorphism groups. We show that our formalization is complete. Secondly, we provide general theoretical evidence of why spring algorithms can display graph symmetry. Finally, the strength of our general theorem is demonstrated from its application to several existing spring algorithms.KeywordsAutomorphism GroupPermutation GroupSpring SystemGraph DrawingGraph LayoutThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In the past decades for more and more graph classes the Graph Isomorphism Problem was shown to be solvable in polynomial time. An interesting family of graph classes arises from intersection graphs of geometric objects. In this work we show that the Graph Isomorphism Problem for unit square graphs, intersection graphs of axis-parallel unit squares in the plane, can be solved in polynomial time. Since the recognition problem for this class of graphs is NP-hard we can not rely on standard techniques for geometric graphs based on constructing a canonical realization. Instead, we develop new techniques which combine structural insights into the class of unit square graphs with understanding of the automorphism group of such graphs. For the latter we introduce a generalization of bounded degree graphs which is used to capture the main structure of unit square graphs. Using group theoretic algorithms we obtain sufficient information to solve the isomorphism problem for unit square graphs.