Graphs Of Rank Research Articles

Spatial realisations of KMS states on the [formula omitted]-algebras of higher-rank graphs

Several authors have recently been studying the equilibrium or KMS states on the Toeplitz algebras of finite higher-rank graphs. For graphs of rank one (that is, for ordinary directed graphs), there is a natural dynamics obtained by lifting the gauge action of the circle to an action of the real line. The algebras of higher-rank graphs carry a gauge action of a higher-dimensional torus, and there are many potential dynamics arising from different embeddings of the real line in the torus. Previous results show that there is nonetheless a “preferred dynamics” for which the system exhibits a particularly satisfactory phase transition, and that the unique KMS state at the critical inverse temperature can then be implemented by integrating vector states against a measure on the infinite path space of the graph. Here we obtain a similar description of the KMS state at the critical inverse temperature for other dynamics. Our spatial implementation is given by integrating against a measure on a space of paths which are infinite in some directions but finite in others. Our results are sharpest for the algebras of rank-two graphs.

Mapping tori of small dilatation expanding train-track maps

An expanding train-track map on a graph of rank n is P-small if its dilatation is bounded above by Pn. We prove that for every P there is a finite list of mapping tori X1,…,XA, with A depending only on P and not n, so that the mapping torus associated with every P-small expanding train-track map can be obtained by surgery on some Xi. We also show that, given an integer P>0, there is a bound M depending only on P and not n, so that the fundamental group of the mapping torus of any P-small expanding train-track map has a presentation with less than M generators and M relations. We also provide some bounds for the smallest possible dilatation.

The accuracy of graphs to describe size distributions

This article analyses the performance of the graphs traditionally used to study size distributions: histograms, Zipf plots (double logarithmic graphs of rank compared to size) and plotted cumulative density functions. A lognormal distribution is fitted to urban data from three countries (the United States, Spain and Italy) over all of the twentieth century. We explain the advantages and disadvantages associated with these graphic methods and derive some statistical properties.

KMS states on [formula omitted]-algebras associated to higher-rank graphs

Consider a higher-rank graph of rank k. Both the Cuntz–Krieger algebra and the Toeplitz–Cuntz–Krieger algebra of the graph carry natural gauge actions of the torus Tk, and restricting these gauge actions to one-parameter subgroups of Tk gives dynamical systems involving actions of the real line. We study the KMS states of these dynamical systems. We find that for large inverse temperatures β, the simplex of KMSβ states on the Toeplitz–Cuntz–Krieger algebra has dimension d one less than the number of vertices in the graph. We also show that there is a preferred dynamics for which there is a critical inverse temperature βc: for β larger than βc, there is a d-dimensional simplex of KMS states; when β=βc and the one-parameter subgroup is dense, there is a unique KMS state, and this state factors through the Cuntz–Krieger algebra. As in previous studies for k=1, our main tool is the Perron–Frobenius theory for irreducible non-negative matrices, though here we need a version of the theory for commuting families of matrices.

Diffusion approximation for the components in critical inhomogeneous random graphs of rank 1.

AbstractConsider the random graph on n vertices 1,…,n. Each vertex i is assigned a type xi with x1,…,xn being independent identically distributed as a nonnegative random variable X. We assume that EX3< ∞. Given types of all vertices, an edge exists between vertices i and j independent of anything else and with probability \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\min \{1, \frac{x_ix_j}{n}\left(1+\frac{a}{n^{1/3}} \right) \}\end{align*} \end{document}. We study the critical phase, which is known to take place when EX2 = 1. We prove that normalized by n‐2/3the asymptotic joint distributions of component sizes of the graph equals the joint distribution of the excursions of a reflecting Brownian motion with diffusion coefficient \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\sqrt{{\textbf{ E}}X{\textbf{ E}}X^3}\end{align*} \end{document}and drift \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}a-\frac{{\textbf{ E}}X^3}{{\textbf{ E}}X}s\end{align*} \end{document}. In particular, we conclude that the size of the largest connected component is of order n2/3. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 43, 486–539, 2013

The diameter of the thick part of moduli space and simultaneous Whitehead moves

Let S be a surface of genus g with p punctures with negative Euler characteristic. We study the diameter of the ϵ-thick part of moduli space of S equipped with the Teichmüller or Thurston’s Lipschitz metric. We show that the asymptotic behaviors in both metrics are of order log(g+pϵ). The same result also holds for the ϵ-thick part of the moduli space of metric graphs of rank n equipped with the Lipschitz metric. The proof involves a sorting algorithm that sorts an arbitrarily labeled tree with n labels using simultaneous Whitehead moves, where the number of steps is of order log(n). As a related combinatorial problem, we also compute, in the appendix of this paper, the asymptotic diameter of the moduli space of pants decompositions on S in the metric of elementary moves.

A characterization of long graphs of arbitrary rank

The rank of a graph G is defined to be the rank of its adjacency matrix. A connected graph G of rank n is called long if it has maximum diameter among all connected graphs with rank n. In this paper, we characterize the long graphs of rank n, for n an arbitrary positive integer.

Vertex rainbow colorings of graphs

In a properly vertex-colored graph G, a path P is a rainbow path if no two vertices of P have the same color, except possibly the two end-vertices of P. If every two vertices of G are connected by a rainbow path, then G is vertex rainbow-connected. A proper vertex coloring of a connected graph G that results in a vertex rainbow-connected graph is a vertex rainbow coloring of G. The minimum number of colors needed in a vertex rainbow coloring of G is the vertex rainbow connection number vrc(G) of G. Thus if G is a connected graph of order n ≥ 2, then 2 ≤ vrc(G) ≤ n. We present characterizations of all connected graphs G of order n for which vrc(G) ∈ {2,n−1,n} and study the relationship between vrc(G) and the chromatic number χ(G) of G. For a connected graph G of order n and size m, the number m − n + 1 is the cycle rank of G. Vertex rainbow connection numbers are determined for all connected graphs of cycle rank 0 or 1 and these numbers are investigated for connected graphs of cycle rank 2.

Affine dual equivalence and $k$-Schur functions

The k-Schur functions were first introduced by Lapointe, Lascoux and Morse (2003) in the hopes of refining the expansion of Macdonald polynomials into Schur functions. Recently, an alternative definition for k-Schur functions was given by Lam, Lapointe, Morse, and Shimozono (2010) as the weighted generating function of starred strong tableaux which correspond with labeled saturated chains in the Bruhat order on the affine symmetric group modulo the symmetric group. This definition has been shown to correspond to the Schubert basis for the affine Grassmannian of type A by Lam (2008), and, at t = 1, it is equivalent to the k-tableaux characterization of Lapointe and Morse (2007). In this paper, we extend Haiman's (1992) dual equivalence relation on standard Young tableaux to all starred strong tableaux. The elementary equivalence relations can be interpreted as labeled edges in a graph which share many of the properties of Assaf's dual equivalence graphs. These graphs display much of the complexity of working with k-Schur functions and the interval structure on affine Symmetric Group modulo the Symmetric Group. We introduce the notions of flattening and squashing skew starred strong tableaux in analogy with jeu da taquin slides in order to give a method to find all isomorphism types for affine dual equivalence graphs of rank 4. Finally, we make connections between k-Schur functions and both LLT and Macdonald polynomials by comparing the graphs for these functions.

On the graph complement conjecture for minimum rank

The minimum rank of a graph has been an interesting and well studied parameter investigated by many researchers over the past decade or so. One of the many unresolved questions on this topic is the so-called graph complement conjecture, which grew out of a workshop in 2006. This conjecture asks for an upper bound on the sum of the minimum rank of a graph and the minimum rank of its complement, and may be classified as a Nordhaus–Gaddum type problem involving the graph parameter minimum rank. The conjectured bound is the order of the graph plus two. Other variants of the graph complement conjecture are introduced here for the minimum semidefinite rank and the Colin de Verdière type parameter ν. We show that if the ν-graph complement conjecture is true for two graphs then it is true for the join of these graphs. Related results for the graph complement conjecture (and the positive semidefinite version) for joins of graphs are also established. We also report on the use of recent results on partial k-trees to establish the graph complement conjecture for graphs of low minimum rank.

On the automorphism group of the Aschbacher graph

A Moore graph is a regular graph of degree k and diameter d with v vertices such that v ≤ 1 + k + k(k − 1) + ... + k(k − 1)d−1. It is known that a Moore graph of degree k ≥ 3 has diameter 2; i.e., it is strongly regular with parameters λ = 0, µ = 1, and v = k 2 + 1, where the degree k is equal to 3, 7, or 57. It is unknown whether there exists a Moore graph of degree k = 57. Aschbacher showed that a Moore graph with k = 57 is not a graph of rank 3. In this connection, we call a Moore graph with k = 57 the Aschbacher graph and investigate its automorphism group G without additional assumptions (earlier, it was assumed that G contains an involution).

On the automorphism group of the Aschbacher graph

A Moore graph is a regular graph of degree k and diameter d with v vertices such that v ≤ 1 + k + k(k − 1) + ... + k(k − 1)d−1. It is known that a Moore graph of degree k ≥ 3 has diameter 2; i.e., it is strongly regular with parameters λ = 0, µ = 1, and v = k 2 + 1, where the degree k is equal to 3, 7, or 57. It is unknown whether there exists a Moore graph of degree k = 57. Aschbacher showed that a Moore graph with k = 57 is not a graph of rank 3. In this connection, we call a Moore graph with k = 57 the Aschbacher graph and investigate its automorphism group G without additional assumptions (earlier, it was assumed that G contains an involution).

Some graphs related to the small Mathieu groups

Two different constructions are given of a rank 8 arc-transitive graph with 165 vertices and valency 8, whose automorphism group is M 11 . One involves 3-subsets of an 11-set while the other involves 4-subsets of a 12-set, and the constructions are linked with the Witt designs on 11, 12 and 24 points. Four different constructions are given of a rank 9 arc-transitive graph with 55 vertices and valency 6 whose automorphism group is PSL ( 2 , 11 ) . This graph occurs as a subgraph of the M 11 graph, and two of the constructions involve 2-subsets of an 11-set while the remaining two involve 3-subsets of an 11-set. The PSL ( 2 , 11 ) and M 11 graphs occur as the second and third members of a tower of graphs defined on a conjugacy class of involutions of the simple groups A 5 , PSL ( 2 , 11 ) , M 11 and M 12 with two involutions adjacent if they generate a special S 3 . The first graph in the tower is the line graph of the Petersen graph while the fourth is the Johnson graph J ( 12 , 4 ) . The graphs also arise as collineation graphs of rank 2 truncations of various residues of certain P -geometries.

The galaxies of nonstandard enlargements of infinite and transfinite graphs

The galaxies of the nonstandard enlargements of connected, conventionally infinite graphs as well as of walk-connected transfinite graphs are defined, analyzed, and illustrated by some examples. It is then shown that any such enlargement either has exactly one galaxy, its principal one, or it has infinitely many galaxies. In the latter case, the galaxies are partially ordered by their “closeness” to the principal galaxy. If an enlargement has a galaxy Γ different from its principal galaxy, then it has a two-way infinite sequence of galaxies that contains Γ and is totally ordered according to that “closeness” property. There may be many such totally ordered sequences. Furthermore, a walk-connected graph G 1 of transfinite rank 1 consists in general of connected conventional graphs (graphs of rank 0, called 0-sections) that are walk-connected together at their infinite extremities. The enlargement ∗ G 1 of G 1 consists of the enlargement of G 1 , as well as of the enlargements of its 0-sections. The latter enlargements are all contained within the principal galaxy of ∗ G 1 . Moreover, ∗ G 1 may have other galaxies of rank 1; these too are partially and totally ordered as before. These results extend to the enlargements of transfinite graphs of ranks greater than 1.

On minimal rank over finite fields

Let F be a field. Given a simple graph G on n vertices, its minimal rank (with respect to F) is the minimum rank of a symmetric n × n F-valued matrix whose off-diagonal zeroes are the same as in the adjacency matrix of G. If F is finite, then for every k, it is shown that the set of graphs of minimal rank at most k is characterized by finitely many forbidden induced subgraphs, each on at most ( |F| k /2 + 1)2 vertices. These findings also hold in a more general context.

The maximum order of finite groups of outer automorphisms of free groups

Let Fr be free group of rank r > l and O u t F r = A u t Fr/InnFr, the outer au tomorph ism group (automorphisms modulo inner automorphisms). We shall prove the following. Theorem. The maximum order of a f ini te subgroup G of Out F, is 12, for r= 2, and 2 r r !, for r > 2. Furthermore the f inite subgroup of Out F, realizing the maximum order is unique up to conjugacy, for r > 3. The proof of the above theorem is based on the following geometric realization result which was observed first in [Z, p. 478], see also [Cu]. Proposition 0 Let G be a .finite subgroup of Out F,. Then there exist a f inite connected graph F with ~ 1 F = F , and an action of G on F realizing the given action on F,. Remarks and definitions. Let F' be a subgraph obtained by deleting all free edges of F, i.e., edges with vertex of valence 1. Note that the G action restricts on the new graph F ' and the fundamental g roup and the induced action on it do not change. So we may assume in Proposi t ion 0 that F contains no vertex of valence 1. We use Sym F to denote the symmetry group of F (in a combina tor ial sense, i.e., mapping vertices to vertices and oriented edges to oriented edges, or equivalently, homeomorph i sms acting linearly on edges). In general we allow inversions of edges, i.e., reflections in the midpoints of edges (which is not considered as the identity). Then, in Proposi t ion 0, we may also assume that F has no vertices of valence 2 by amalgamat ing the two adjacent edges containing the same vertex of valence 2 into one edge, again G acts on the new graph with the same induced action on the fundamenta l group. We say that a g roup G acts effectively on F if it can be embedded into Sym F, or equivalently, no non-trivial element of G is the identity on F. For a connected graph F, its rank, denoted by rank F, is the rank of its fundamental group. Lemma 1 Let F be a connected graph of rank r > 1 without vertices of valence 1 and G be a f ini te group acting on F. Then G acts effectively on F if and only if the induced action on ~ F injects into Out ~ F.