scholarly journals Vertex-colored graphs, bicycle spaces and Mahler measure

2016 ◽  
Vol 25 (06) ◽  
pp. 1650033 ◽  
Author(s):  
Kalyn R. Lamey ◽  
Daniel S. Silver ◽  
Susan G. Williams
Keyword(s):  
Polynomial Ring ◽  
Spanning Trees ◽  
Finite Graph ◽  
Mahler Measure ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Locally Finite ◽  
Polynomial Invariants ◽  
Colored Graphs ◽  
Locally Finite Graph

The space [Formula: see text] of conservative vertex colorings (over a field [Formula: see text]) of a countable, locally finite graph [Formula: see text] is introduced. When [Formula: see text] is connected, the subspace [Formula: see text] of based colorings is shown to be isomorphic to the bicycle space of the graph. For graphs [Formula: see text] with a cofinite free [Formula: see text]-action by automorphisms, [Formula: see text] is dual to a finitely generated module over the polynomial ring [Formula: see text]. Polynomial invariants for this module, the Laplacian polynomials [Formula: see text], are defined, and their properties are discussed. The logarithmic Mahler measure of [Formula: see text] is characterized in terms of the growth of spanning trees.

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 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

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).

scholarly journals Relating Different Cycle Spaces of the Same Infinite Graph

10.37236/622 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
R. Bruce Richter ◽  
Brendan Rooney
Keyword(s):  
Finite Graph ◽  
Infinite Graph ◽  
Cycle Space ◽  
Locally Finite ◽  
Locally Finite Graph ◽  
Continuous Surjection

Casteels and Richter have shown that if $X$ and $Y$ are distinct compactifications of a locally finite graph $G$ and $f:X\to Y$ is a continuous surjection such that $f$ restricts to a homeomorphism on $G$, then the cycle space $Z_X$ of $X$ is contained in the cycle space $Z_Y$ of $Y$. In this work, we show how to extend a basis for $Z_X$ to a basis of $Z_Y$.

scholarly journals Geodetic Topological Cycles in Locally Finite Graphs

10.37236/233 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Agelos Georgakopoulos ◽  
Philipp Sprüssel
Keyword(s):  
Finite Graph ◽  
Cycle Space ◽  
Locally Finite ◽  
Finite Graphs ◽  
Locally Finite Graph ◽  
Locally Finite Graphs

We prove that the topological cycle space ${\cal C}(G)$ of a locally finite graph $G$ is generated by its geodetic topological circles. We further show that, although the finite cycles of $G$ generate ${\cal C}(G)$, its finite geodetic cycles need not generate ${\cal C}(G)$.

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
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