The finiteness of the number of symmetrical extensions of a locally finite tree by a finite graph

2016 ◽  
Vol 295 (S1) ◽  
pp. 168-173
Author(s):  
V. I. Trofimov
Keyword(s):  
Finite Graph ◽  
Finite Tree ◽  
Locally Finite

scholarly journals CONTEXT-FREE GROUPS AND THEIR STRUCTURE TREES

2013 ◽  
Vol 23 (03) ◽  
pp. 611-642 ◽  
Author(s):  
VOLKER DIEKERT ◽  
ARMIN WEIß
Keyword(s):  
Free Groups ◽  
Finite Graph ◽  
Fundamental Result ◽  
Direct Construction ◽  
Finite Tree ◽  
Locally Finite ◽  
Locally Finite Graph ◽  
Structure Tree ◽  
Tree Width ◽  
Context Free

Let Γ be a connected, locally finite graph of finite tree width and G be a group acting on it with finitely many orbits and finite node stabilizers. We provide an elementary and direct construction of a tree T on which G acts with finitely many orbits and finite vertex stabilizers. Moreover, the tree is defined directly in terms of the structure tree of optimally nested cuts of Γ. Once the tree is constructed, Bass–Serre theory yields that G is virtually free. This approach simplifies the existing proofs for the fundamental result of Muller and Schupp that characterizes context-free groups as finitely generated (f.g.) virtually free groups. Our construction avoids the explicit use of Stallings' structure theorem and it is self-contained. We also give a simplified proof for an important consequence of the structure tree theory by Dicks and Dunwoody which has been stated by Thomassen and Woess. It says that a f.g. group is accessible if and only if its Cayley graph is accessible.

scholarly journals Distinguishability of Locally Finite Trees

10.37236/947 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Mark E. Watkins ◽  
Xiangqian Zhou
Keyword(s):  
Positive Integer ◽  
Chromatic Number ◽  
Inverse Image ◽  
Vertex Coloring ◽  
Finite Tree ◽  
Locally Finite ◽  
Distinguishing Number ◽  
Nonidentity Element ◽  
Locally Countable ◽  
Proper Vertex

The distinguishing number $\Delta(X)$ of a graph $X$ is the least positive integer $n$ for which there exists a function $f:V(X)\to\{0,1,2,\cdots,n-1\}$ such that no nonidentity element of $\hbox{Aut}(X)$ fixes (setwise) every inverse image $f^{-1}(k)$, $k\in\{0,1,2,\cdots,n-1\}$. All infinite, locally finite trees without pendant vertices are shown to be 2-distinguishable. A proof is indicated that extends 2-distinguishability to locally countable trees without pendant vertices. It is shown that every infinite, locally finite tree $T$ with finite distinguishing number contains a finite subtree $J$ such that $\Delta(J)=\Delta(T)$. Analogous results are obtained for the distinguishing chromatic number, namely the least positive integer $n$ such that the function $f$ is also a proper vertex-coloring.

scholarly journals Lower Bounds for Identifying Codes in some Infinite Grids

10.37236/394 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Ryan Martin ◽  
Brendon Stanton
Keyword(s):  
Lower Bounds ◽  
Finite Graph ◽  
Infinite Graphs ◽  
Locally Finite ◽  
Identifying Codes ◽  
Identifying Code ◽  
Hexagonal Grids ◽  
Closed Neighborhood ◽  
Definition Of ◽  
Infinite Grids

An $r$-identifying code on a graph $G$ is a set $C\subset V(G)$ such that for every vertex in $V(G)$, the intersection of the radius-$r$ closed neighborhood with $C$ is nonempty and unique. On a finite graph, the density of a code is $|C|/|V(G)|$, which naturally extends to a definition of density in certain infinite graphs which are locally finite. We present new lower bounds for densities of codes for some small values of $r$ in both the square and hexagonal grids.

Asymptotic Behavior of Skew Conditional Heat Kernels on Graph Networks

1993 ◽  
Vol 45 (4) ◽  
pp. 863-878 ◽  
Author(s):  
Tatsuya Okada
Keyword(s):  
Asymptotic Behavior ◽  
Finite Graph ◽  
Heat Propagation ◽  
Heat Kernels ◽  
Locally Finite ◽  
Locally Finite Graph ◽  
Periodic Models ◽  
Definition Of

AbstractIn this note, we will consider the heat propagation on locally finite graph networks which satisfy a skew condition on vertices (See Definition of Section 2). For several periodic models, we will construct the heat kernels Pt with the skew condition explicitly, and derive the decay order of Pt as time goes to infinity.

scholarly journals Discrete Schrödinger operators on a graph

1992 ◽  
Vol 125 ◽  
pp. 141-150 ◽  
Author(s):  
Polly Wee Sy ◽  
Toshikazu Sunada
Keyword(s):  
Riemannian Manifold ◽  
Automorphism Group ◽  
Connected Graph ◽  
Spectral Properties ◽  
Orbit Space ◽  
Finite Graph ◽  
Discrete Analogue ◽  
Discrete Laplacian ◽  
Locally Finite ◽  
Discrete Schrödinger Operators

In this paper, we study some spectral properties of the discrete Schrödinger operator = Δ + q defined on a locally finite connected graph with an automorphism group whose orbit space is a finite graph.The discrete Laplacian and its generalization have been explored from many different viewpoints (for instance, see [2] [4]). Our paper discusses the discrete analogue of the results on the bottom of the spectrum established by T. Kobayashi, K. Ono and T. Sunada [3] in the Riemannian-manifold-setting.

scholarly journals The homology of a locally finite graph with ends

COMBINATORICA ◽  
2010 ◽  
Vol 30 (6) ◽  
pp. 681-714 ◽  
Author(s):  
Reinhard Diestel ◽  
Philipp Sprüssel
Keyword(s):  
Finite Graph ◽  
Locally Finite ◽  
Locally Finite Graph

Matchings in Arbitrary Graphs

1971 ◽  
Vol 69 (3) ◽  
pp. 401-407 ◽  
Author(s):  
R. A. Brualdi
Keyword(s):  
Perfect Matching ◽  
Sufficient Conditions ◽  
Finite Graph ◽  
Maximum Cardinality ◽  
Original Proof ◽  
Locally Finite ◽  
Finite Graphs ◽  
Locally Finite Graphs ◽  
Arbitrary Graphs ◽  
Necessary And Sufficient

1. Tutte(10) has given necessary and sufficient conditions in order that a finite graph have a perfect matching. A different proof was given by Gallai(4). Berge(1) (and Ore (7)) generalized Tutte's result by determining the maximum cardinality of a matching in a finite graph. In his original proof Tutte used the method of skew symmetric determinants (or pfaffians) while Gallai and Berge used the much exploited method of alternating paths. Another proof of Berge's theorem, along with an efficient algorithm for constructing a matching of maximum cardinality, was given by Edmonds (2). In another paper (12) Tutte extended his conditions for a perfect matching to locally finite graphs.

scholarly journals The Planarity Theorems of MacLane and Whitney for Graph-like Continua

10.37236/284 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Robin Christian ◽  
R. Bruce Richter ◽  
Brendan Rooney
Keyword(s):  
Finite Graph ◽  
Disjoint Paths ◽  
Locally Finite ◽  
Edge Disjoint ◽  
Locally Finite Graph ◽  
Compact Graph

The planarity theorems of MacLane and Whitney are extended to compact graph-like spaces. This generalizes recent results of Bruhn and Stein (MacLane's Theorem for the Freudenthal compactification of a locally finite graph) and of Bruhn and Diestel (Whitney's Theorem for an identification space obtained from a graph in which no two vertices are joined by infinitely many edge-disjoint paths).

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