A finite graph is homeomorphic to the Reeb graph of a Morse–Bott function

2021 ◽  
Vol 71 (3) ◽  
pp. 757-772
Author(s):  
Irina Gelbukh
Keyword(s):  
Fundamental Group ◽  
Height Function ◽  
Finite Graph ◽  
Critical Values ◽  
Reeb Graph ◽  
Special Cases ◽  
Cycle Rank ◽  
Multiple Edges

Abstract We prove that a finite graph (allowing loops and multiple edges) is homeomorphic (isomorphic up to vertices of degree two) to the Reeb graph of a Morse–Bott function on a smooth closed n-manifold, for any dimension n ≥ 2. The manifold can be chosen orientable or non-orientable; we estimate the co-rank of its fundamental group (or the genus in the case of surfaces) from below in terms of the cycle rank of the graph. The function can be chosen with any number k ≥ 3 of critical values, and in a few special cases with k < 3. In the case of surfaces, the function can be chosen, except for a few special cases, as the height function associated with an immersion ℝ3.

scholarly journals Approximation of metric spaces by Reeb graphs: Cycle rank of a Reeb graph, the co-rank of the fundamental group, and large components of level sets on Riemannian manifolds

Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 2031-2049
Author(s):  
Irina Gelbukh
Keyword(s):  
Riemannian Manifold ◽  
Continuous Function ◽  
Fundamental Group ◽  
Metric Spaces ◽  
Reeb Graph ◽  
Topological Graph ◽  
Reeb Graphs ◽  
Local Path ◽  
Cycle Rank ◽  
Euclidean Unit Ball

For a connected locally path-connected topological space X and a continuous function f on it such that its Reeb graph Rf is a finite topological graph, we show that the cycle rank of Rf, i.e., the first Betti number b1(Rf), in computational geometry called number of loops, is bounded from above by the co-rank of the fundamental group ?1(X), the condition of local path-connectedness being important since generally b1(Rf) can even exceed b1(X). We give some practical methods for calculating the co-rank of ?1(X) and a closely related value, the isotropy index. We apply our bound to improve upper bounds on the distortion of the Reeb quotient map, and thus on the Gromov-Hausdorff approximation of the space by Reeb graphs, for the distance function on a compact geodesic space and for a simple Morse function on a closed Riemannian manifold. This distortion is bounded from below by what we call the Reeb width b(M) of a metric space M, which guarantees that any real-valued continuous function on M has large enough contour (connected component of a level set). We show that for a Riemannian manifold, b(M) is non-zero and give a lower bound on it in terms of characteristics of the manifold. In particular, we show that any real-valued continuous function on a closed Euclidean unit ball E of dimension at least two has a contour C with diam(C??E)??3.

Graphs with 6-Ways

1973 ◽  
Vol 25 (4) ◽  
pp. 687-692 ◽  
Author(s):  
John L. Leonard
Keyword(s):  
Finite Graph ◽  
Minimum Number ◽  
Multiple Edges

In a finite graph with no loops nor multiple edges, two points a and b are said to be connected by an r-way, or more explicitly, by a line r-way a — b if there are r paths, no two of which have lines in common (although they may share common points), which join a to b. In this note we demonstrate that any graph with n points and 3n — 2 or more lines must contain a pair of points joined by a 6-way, and that 3n — 2 is the minimum number of lines which guarantees the presence of a 6-way in a graph of n points.In the language of [3], this minimum number of lines needed to guarantee a 6-way is denoted U(n). For the background of this problem, the reader is referred to [3].

scholarly journals Enumerating Tree-Like Graphs and Polymer Topologies with a Given Cycle Rank

Entropy ◽  
2020 ◽  
Vol 22 (11) ◽  
pp. 1295
Author(s):  
Naveed Ahmed Azam ◽  
Aleksandar Shurbevski ◽  
Hiroshi Nagamochi
Keyword(s):  
Chemical Compounds ◽  
Canonical Representation ◽  
Cycle Rank ◽  
Multiple Edges ◽  
Isomorphic Graphs ◽  
Important Notion

Cycle rank is an important notion that is widely used to classify, understand, and discover new chemical compounds. We propose a method to enumerate all non-isomorphic tree-like graphs of a given cycle rank with self-loops and no multiple edges. To achieve this, we develop an algorithm to enumerate all non-isomorphic rooted graphs with the required constraints. The idea of our method is to define a canonical representation of rooted graphs and enumerate all non-isomorphic graphs by generating the canonical representation of rooted graphs. An important feature of our method is that for an integer n≥1, it generates all required graphs with n vertices in O(n) time per graph and O(n) space in total, without generating invalid intermediate structures. We performed some experiments to enumerate graphs with a given cycle rank from which it is evident that our method is efficient. As an application of our method, we can generate tree-like polymer topologies of a given cycle rank with self-loops and no multiple edges.

scholarly journals Isomorphic subgraphs having minimal intersections

1983 ◽  
Vol 35 (3) ◽  
pp. 287-306
Author(s):  
R. C. Mullin ◽  
B. K. Roy ◽  
P. J. Schellenberg
Keyword(s):  
Complete Graph ◽  
Upper Bound ◽  
Packing Problem ◽  
Finite Graph ◽  
Combinatorial Problems ◽  
Set Packing ◽  
Complete Subgraph ◽  
Special Cases ◽  
Set Packing Problem ◽  
Special Classes

AbstractGiven a finite graph H and G, a subgraph of it, we define σ (G, H) to be the largest integer such that every pair of subgraphs of H, both isomorphic to G, has at least σ(G, H) edges in common; furthermore, R(G, H) is defined to be the maximum number of subgraphs of H, all isomorphic to G, such that any two of them have σ(G, H) edges common between them. We are interested in the values of σ(G, H) and R(G, H) for general H and G. A number of combinatorial problems can be considered as special cases of this question; for example, the classical set-packing problem is equivalent to evaluating R (G, H) where G is a complete subgraph of the complete graph H and σ(G, H) = 0, and the decomposition of H into subgraphs isomorphic to G is equivalent to showing that σ(G, H) = 0 and R(G, H) = ε(H)/ε(G) where ε(H), ε(G) are the number of edges in H, G respectively.A result of S. M. Johnson (1962) gives an upper bound for R(G, H) in terms of σ(G, H). As a corollary of Johnson's result, we obtain the upper bound of McCarthy and van Rees (1977) for the Cordes problem. The remainder of the paper is a study of σ (G, H) and R(G, H) for special classes of graphs; in particular, H is a complete graph and G is, in most instances, a union of disjoint complete subgraphs.

scholarly journals RELATIVE TORSION

2001 ◽  
Vol 03 (01) ◽  
pp. 15-85 ◽  
Author(s):  
DAN BURGHELEA ◽  
LEONID FRIEDLANDER ◽  
THOMAS KAPPELER
Keyword(s):  
Fundamental Group ◽  
Finite Type ◽  
Von Neumann Algebra ◽  
Hilbert Module ◽  
Von Neumann ◽  
Geometric Invariants ◽  
Finite Dimensional ◽  
Finite Von Neumann Algebra ◽  
Special Cases ◽  
Relative Torsion

This paper achieves, among other things, the following: • It frees the main result of [9] from the hypothesis of determinant class and extends this result from unitary to arbitrary representations. • It extends (and at the same times provides a new proof of) the main result of Bismut and Zhang [3] from finite dimensional representations of Γ to representations on an [Formula: see text]-Hilbert module of finite type ([Formula: see text] a finite von Neumann algebra). The result of [3] corresponds to [Formula: see text]. • It provides interesting real valued functions on the space of representations of the fundamental group Γ of a closed manifold M. These functions might be a useful source of topological and geometric invariants of M. These objectives are achieved with the help of the relative torsion ℛ, first introduced by Carey, Mathai and Mishchenko [12] in special cases. The main result of this paper calculates explicitly this relative torsion (cf. Theorem 1.1).

scholarly journals The fundamental group of a locally finite graph with ends

2011 ◽  
Vol 226 (3) ◽  
pp. 2643-2675 ◽  
Author(s):  
Reinhard Diestel ◽  
Philipp Sprüssel
Keyword(s):  
Fundamental Group ◽  
Finite Graph ◽  
Locally Finite ◽  
Locally Finite Graph

Curvature-torsion quasitensor of Laptev fundamental-group connection

2020 ◽  
pp. 155-169
Author(s):  
Yu. I. Shevchenko
Keyword(s):  
Fundamental Group ◽  
Curvature Tensor ◽  
Principal Bundle ◽  
Torsion Tensor ◽  
Structural Equations ◽  
The Other ◽  
Cartan Connection ◽  
Cartan Connections ◽  
Special Cases ◽  
The Given

We consider a space with Laptev's fundamental group connection generalizing spaces with Cartan connections. Laptev structural equations are reduced to a simpler form. The continuation of the given structural equations made it possible to find differential comparisons for the coeffi­cients in these equations. It is proved that one part of these coefficients forms a tensor, and the other part forms is quasitensor, which justifies the name quasitensor of torsion-curvature for the entire set. From differential congruences for the components of this quasitensor, congruences are ob­tained for the components of the Laptev curvature-torsion tensor, which contains 9 subtensors included in the unreduced structural equations. In two special cases, a space with a fundamental connection is a spa­ce with a Cartan connection, having a quasitensor of torsion-curvature, which contains a quasitensor of torsion. In the reductive case, the space of the Cartan connection is turned into such a principal bundle with connec­tion that has not only a curvature tensor, but also a torsion tensor.

scholarly journals A note on the conditional chromatic polynomial

1994 ◽  
Vol 36 (3) ◽  
pp. 265-267 ◽  
Author(s):  
V. Voloshin
Keyword(s):  
Finite Graph ◽  
Chromatic Polynomial ◽  
Multiple Edges

In this note we consider a finite graph without loops and multiple edges. The colouring of a graph G in λ colours is the colouring of its vertices in such a way that no two of adjacent vertices have the same colours and the number of used colours does not exceed λ [1, 4]. Two colourings of graph G are called different if there exists at least one vertex which changes colour when passing from one colouring to another.

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