Graph Theory

Diestel, Reinhard

doi:10.1007/978-3-662-53622-3

Cited by 679 publications

(371 citation statements)

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“…An oriented graph is a directed graph with no bidirected edges. The adjacency matrix of an oriented graph is non-Hermitian [9]. This means that the recipe outlined in the previous section does not work because the exponentiation of a non-Hermitian operator generates a non-unitary operator.…”

Section: Oriented Graphsmentioning

confidence: 99%

Discrete-Time Quantum Walks on Oriented Graphs

Chagas

Portugal²

2020

Electron. Proc. Theor. Comput. Sci.

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The interest in quantum walks has been steadily increasing during the last two decades. It is still worth to present new forms of quantum walks that might find practical applications and new physical behaviors. In this work, we define discrete-time quantum walks on arbitrary oriented graphs by partitioning a graph into tessellations, which is a collection of disjoint cliques that cover the vertex set. By using the adjacency matrices associated with the tessellations, we define local unitary operators, whose product is the evolution operator of our quantum walk model. We introduce a parameter, called α, that quantifies the amount of orientation. We show that the parameter α can be tuned in order to increase the amount of quantum walk-based transport on oriented graphs.

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Section: Oriented Graphsmentioning

confidence: 99%

Discrete-Time Quantum Walks on Oriented Graphs

Chagas

Portugal²

2020

Electron. Proc. Theor. Comput. Sci.

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“…Let , , and denote the complete graph, the path and the cycle of order , respectively. For terms and symbols not defined here, we refer the reader to [3].…”

Section: A M S 2 0 1 0 M a T H E M A T I C S S U B J E C T C L A S S mentioning

confidence: 99%

“…Let , , and denote the complete graph, the path and the cycle of order , respectively. For terms and symbols not defined here, we refer the reader to [3].Let again be a graph. A subset of ( ) is a matching if no two distinct edges in have a common endvertex.…”

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confidence: 99%

Sufficient conditions for the existence of a path‐factor which are related to odd components

Egawa

Furuya

Ozeki

2018

Journal of Graph Theory

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In this article, we are concerned with sufficient conditions for the existence of a {P2,P2k+1}‐factor. We prove that for k≥3, there exists εk>0 such that if a graph G satisfies 0true∑0≤j≤k−1c2j+1(G−X)≤εk|X| for all X⊆V(G), then G has a {P2,P2k+1}‐factor, where cifalse(G−Xfalse) is the number of components C of G−X with |V(C)|=i. On the other hand, we construct infinitely many graphs G having no {P2,P2k+1}‐factor such that 0true∑0≤j≤k−1c2j+1(G−X)≤32k+14172k−78|X| for all X⊆V(G).

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“…For a connected graph

F

G

is said to be

F

‐ free if

G

contains no

F

as an induced subgraph. For terms and symbols not defined in this paper, we refer the readers to .…”

Section: Introductionmentioning

confidence: 99%

Large homeomorphically irreducible trees in path‐free graphs

Furuya

Tsuchiya

2019

Journal of Graph Theory

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A subgraph of a graph is a homeomorphically irreducible tree (HIT) if it is a tree with no vertices of degree two. Furuya and Tsuchiya [Discrete Math. 313 (2013), pp. 2206–2212] proved that every connected P4‐free graph of order ngoodbreakinfix≥5 has a HIT of order at least ngoodbreakinfix−1, and Diemunsch et al [Discrete Appl. Math. 185 (2015), pp. 71–78] proved that every connected P5‐free graph of order ngoodbreakinfix≥6 has a HIT of order at least ngoodbreakinfix−2. In this paper, we continue the study of HITs in path‐free graphs and bring to a conclusion proving the following three results: (a) If a connected graph F satisfies the condition that every connected F‐free graph G has a HIT of order normalΩ(∣V(G)∣), then F is a path of order at most 7. (b) Every connected P6‐free graph of order ngoodbreakinfix≥9 has a HIT of order at least (n+1)/2. (c) Every connected P7‐free graph of order ngoodbreakinfix≥9 has a HIT of order at least (n+3)/3. Furthermore, we focus on a P6‐free graph having large minimum degree, and prove that for a connected P6‐free graph G of order ngoodbreakinfix≥11, if δ(G)goodbreakinfix≥3, then G has a HIT of order greater than (δ(G)n/δ(G)+1)−1.

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Graph Theory

Cited by 679 publications

References 0 publications

Discrete-Time Quantum Walks on Oriented Graphs

Discrete-Time Quantum Walks on Oriented Graphs

Sufficient conditions for the existence of a path‐factor which are related to odd components

Large homeomorphically irreducible trees in path‐free graphs

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