On Vector Spaces Associated with a Graph

Chen, Wai‐Kai

doi:10.1137/0120054

Cited by 47 publications

(25 citation statements)

References 2 publications

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“…X). According to a theorem of Chen [Che71] (see also [Big97, Proposition 35.2]), G has an Eulerian cut if and only if κ G is even. From Corollary 4.15, it therefore follows that if G ′ has an Eulerian cut and there exists a non-constant harmonic morphism from G to G ′ , then G has an Eulerian cut as well.…”

Section: Functorial Propertiesmentioning

confidence: 99%

Harmonic Morphisms and Hyperelliptic Graphs

Baker

Norine

2009

International Mathematics Research Notices

166

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Abstract. We study harmonic morphisms of graphs as a natural discrete analogue of holomorphic maps between Riemann surfaces. We formulate a graph-theoretic analogue of the classical RiemannHurwitz formula, study the functorial maps on Jacobians and harmonic 1-forms induced by a harmonic morphism, and present a discrete analogue of the canonical map from a Riemann surface to projective space. We also discuss several equivalent formulations of the notion of a hyperelliptic graph, all motivated by the classical theory of Riemann surfaces. As an application of our results, we show that for a 2-edge-connected graph G which is not a cycle, there is at most one involution ι on G for which the quotient G/ι is a tree. We also show that the number of spanning trees in a graph G is even if and only if G admits a non-constant harmonic morphism to the graph B2 consisting of 2 vertices connected by 2 edges. Finally, we use the Riemann-Hurwitz formula and our results on hyperelliptic graphs to classify all hyperelliptic graphs having no Weierstrass points.

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Section: Functorial Propertiesmentioning

confidence: 99%

Harmonic Morphisms and Hyperelliptic Graphs

Baker

Norine

2009

International Mathematics Research Notices

166

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“…Here we represent cycles by their edge sets. The set C of Eulerian subgraphs (unions of edge-disjoint cycles) forms a vector space over GF (2) with vector addition X È Y :¼ ðX [ YÞnðX \ YÞ and scalar multiplication 1 Á X ¼ X; 0 Á X ¼ ;; which is called the cycle space of G [1]. The dimension of the cycle space is the cyclomatic number ðGÞ ¼ jEj À jVj þ p; where p is the number of connected components of G.…”

Section: Introductionmentioning

confidence: 99%

Minimum cycle bases of Halin graphs

Stadler

2003

Journal of Graph Theory

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“…[2]. As every 2-connected graph G is connected, the dimension dim GF(2) C(G) of its cycle space coincides with its cyclomatic number µ(G) := |E| − |V | + 1, see e.g.…”

Section: Preliminariesmentioning

confidence: 99%

“…Cycle bases [2,9] are not only an interesting characterization of the structure of graphs by themselves but also provide a basis for computational assessments of the cycle structure of a graph. "Cycle-space algorithms", for instance, attempt to construct the set of all elementary cycles of a graph from a cycle basis B by iteratively computing the symmetric difference of an elementary cycle and a basis cycle, subsequently retaining the result if and only if it is again an elementary cycle.…”

Section: Introductionmentioning

confidence: 99%

A note on quasi-robust cycle bases

et al. 2009

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We investigate here some aspects of cycle bases of undirected graphs that allow the iterative construction of all elementary cycles. We introduce the concept of quasi-robust bases as a generalization of the notion of robust bases and demonstrate that a certain class of bases of the complete bipartite graphs K m,n with m, n ≥ 5 is quasi-robust but not robust. We furthermore disprove a conjecture for cycle bases of Cartesian product graphs.

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On Vector Spaces Associated with a Graph

Cited by 47 publications

References 2 publications

Harmonic Morphisms and Hyperelliptic Graphs

Harmonic Morphisms and Hyperelliptic Graphs

Minimum cycle bases of Halin graphs

A note on quasi-robust cycle bases

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