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### A Note on Circuit Graphs

Qing Cui
<span title="2010-01-31">2010</span> <i title="The Electronic Journal of Combinatorics"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/v5dyak6ulffqfara7hmuchh24a" style="color: black;">Electronic Journal of Combinatorics</a> </i> &nbsp;
We give a short proof of Gao and Richter's theorem that every circuit graph contains a closed walk visiting each vertex once or twice.  ...  In [6] , the authors asked for a result for the number of vertices visited twice of closed 2-walks in circuit graphs or in 3-connected planar graphs, similarly to Theorem 3 for 3-trees. the electronic  ...  Now W := (W * − v * ) + {v, w, v ′ v, vw, ww ′ } gives the desired closed 2-walk in G. Therefore, we may assume that (G * , C * ) is not a circuit graph.  ...
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### Efficient quantum circuit implementation of quantum walks [article]

B. L. Douglas, J. B. Wang
<span title="2009-10-29">2009</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We provide examples of some highly symmetric graphs on which efficient quantum circuits implementing quantum walks can be constructed, and discuss potential applications to quantum search for marked vertices  ...  along these graphs.  ...  Flamia for use of their Mathematica code, which was adapted to construct the toroidal graph figure used here.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0706.0304v3">arXiv:0706.0304v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5ka36mdberepvb7wsj5yelcesy">fatcat:5ka36mdberepvb7wsj5yelcesy</a> </span>
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### Complexity analysis of quantum walk based search algorithms [article]

B. L. Douglas, J. B. Wang
<span title="2014-08-07">2014</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
For these graphs, we construct quantum circuits that explicitly implement the full quantum walk search algorithm, without reference to a 'black box' oracle.  ...  We also provide a numerical analysis of a quantum walk based search along a twisted toroid family of graphs, which requires O(√(n) log(n)) elementary 2-qubit quantum gate operations to find a marked node  ...  Hence as in [2] , after O( √ N) steps of the walk (or equivalently O( √ N) repetitions of the circuit of Fig. 2 , each requiring O(log(N)) elementary quantum gates), the walk will be found at the marked  ...
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### Efficient quantum circuits for Szegedy quantum walks

T. Loke, J.B. Wang
<span title="">2017</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/sz2pf5vm65bf3okvxa5mtm4xhu" style="color: black;">Annals of Physics</a> </i> &nbsp;
In addition, we apply this scheme to construct efficient quantum circuits simulating the Szegedy walks used in the quantum Pagerank algorithm for some classes of non-trivial graphs, providing a necessary  ...  A major advantage in using Szegedy's formalism over discrete-time and continuous-time quantum walks lies in its ability to define a unitary quantum walk on directed and weighted graphs.  ...  Figure 9 : 9 Quantum circuit implementing U walk for the K 2 n 1 ,2 n 2 graph.  ...
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### Quadratic diameter bounds for dual network flow polyhedra [article]

Steffen Borgwardt, Elisabeth Finhold, Raymond Hemmecke
<span title="2014-08-19">2014</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
combinatorial diameter, and |V|· (|V|-1)/2 for the circuit diameter.  ...  In particular, we construct a family of dual network flow polyhedra with members that violate the circuit diameter bound for bipartite graphs by an arbitrary additive constant.  ...  Hence, every edge walk (circuit walk) in P G ′ ,c ′ admits an edge walk (circuit walk) in P G,c that keeps the edge ab, such that we can continue the walk in the smaller polyhedron.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1408.4184v1">arXiv:1408.4184v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5fswmhonlfborjagqqxol5lxsi">fatcat:5fswmhonlfborjagqqxol5lxsi</a> </span>
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### The Circuit Diameter of the Klee-Walkup Polyhedron

Tamon Stephen, Timothy Yusun
<span title="">2015</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/fhi2xwpnh5gmlgof2idwu5wlgq" style="color: black;">Electronic Notes in Discrete Mathematics</a> </i> &nbsp;
Consider a variant of the graph diameter of a polyhedron where each step in a walk between two vertices travels maximally in a circuit direction instead of along incident edges.  ...  It is appealing to consider a circuit analogue of the Hirsch conjecture for graph diameter, as suggested by Borgwardt et al. [2] .  ...  Note that this walk used actual edge directions, and so is more restrictive than circuit directions.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.endm.2015.06.070">doi:10.1016/j.endm.2015.06.070</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/igp77mgz7bbs5da7yujrr462la">fatcat:igp77mgz7bbs5da7yujrr462la</a> </span>
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### Circuit diameter and Klee-Walkup constructions [article]

Tamon Stephen, Timothy Yusun
<span title="2015-03-18">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Consider a variant of the graph diameter of a polyhedron where each step in a walk between two vertices travels maximally in a circuit direction instead of along incident edges.  ...  It is appealing to consider a circuit analogue of the Hirsch conjecture for graph diameter, as suggested by Borgwardt et al. [BFH15].  ...  We then enumerate exhaustively all circuit walks of length 2 or 3 emanating from any point, by considering all possible triples of circuit directions.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1503.05252v1">arXiv:1503.05252v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wug6grx7cragxcjs2jge4bof5m">fatcat:wug6grx7cragxcjs2jge4bof5m</a> </span>
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### A combinatorial approach to counting primitive periodic and primitive pseudo orbits on circulant graphs [article]

Lauren Engelthaler, Isaac Hellerman, Tori Hudgins
<span title="2021-09-28">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
primitive pseudo orbits (sets of distinct primitive periodic orbits) of length up to at least n that lack self-intersections, or that self-intersect only at individual vertices repeated exactly twice (2-  ...  We then regard these two families of graphs as families of quantum graphs and use the counting results to compute the variance of the coefficients of the quantum graph's characteristic polynomial.  ...  the use of walk sums, may be most applicable in the context of circulant graphs. n = 5 l |P l 0 | |a l | 2 Numerics Figure 1 : 1 The directed circulant graph C + 7 Lemma 2. 3 3 On a directed graph G,  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2107.13051v2">arXiv:2107.13051v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hssnvpxo3rbdfadmlaxf6i4qae">fatcat:hssnvpxo3rbdfadmlaxf6i4qae</a> </span>
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### Balanced graphs and noncovering graphs

Oliver Pretzel, Dale Youngs
<span title="">1991</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
In this paper we describe a simple method of constructing a large family of such graphs.  ...  We first construct graphs that have very restricted diagram orientations and then show that identifying certain sets of vertices in one of these graphs produces a noncovering graph. 0012-365X/91/\$03.50  ...  If we ensure that the walks W{ in Theorem 2 reduce to circuits in G' then we can restrict the values f;: in condition (ii) further, because the flow difference of a circuit K in a diagram orientation R  ...
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### Quantum Computing: Implementing Hitting Time for Coined Quantum Walks on Regular Graphs [article]

<span title="2021-08-05">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this paper, we design a quantum circuit for the MNRS algorithm, which finds a marked node in a graph with a quantum walk, and use it to find a hitting time for the marked nodes in the walk.  ...  In recent years, quantum walks have been widely researched and have shown exciting properties. One such is a quadratic speed-up in hitting time compared to its classical counterpart.  ...  We also give quantum circuits for walks with Grover coin on a hypercube, 2-dimensional lattice, complete bipartite graph, and complete graph, which are the graphs that we use in the quantum walk search  ...
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### Randomness-Efficient Sampling Within NC 1 [chapter]

Alexander Healy
<span title="">2006</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Our sampler matches the parameters achieved by random walks on constant-degree expander graphs, allowing us to apply a variety expander-based techniques within N C 1 .  ...  We construct a randomness-efficient averaging sampler that is computable by uniform constantdepth circuits with parity gates (i.e., in uniform AC 0 [⊕]).  ...  of time that a k-step random walk spends in S is 1/2 ± with probability 1 − 2 −Ω( 2 k) .  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/11830924_37">doi:10.1007/11830924_37</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/22ftgu5oqnf5xm6yy6ex2dbfue">fatcat:22ftgu5oqnf5xm6yy6ex2dbfue</a> </span>
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### Randomness-Efficient Sampling within NC1

Alexander D. Healy
<span title="2008-03-24">2008</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/t46w5lpc3ngnfmzil33zdjjrp4" style="color: black;">Computational Complexity</a> </i> &nbsp;
Our sampler matches the parameters achieved by random walks on constant-degree expander graphs, allowing us to apply a variety expander-based techniques within N C 1 .  ...  We construct a randomness-efficient averaging sampler that is computable by uniform constantdepth circuits with parity gates (i.e., in uniform AC 0 [⊕]).  ...  of time that a k-step random walk spends in S is 1/2 ± with probability 1 − 2 −Ω( 2 k) .  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00037-007-0238-5">doi:10.1007/s00037-007-0238-5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6ncbtks35nccvgbzkxhlryyevq">fatcat:6ncbtks35nccvgbzkxhlryyevq</a> </span>
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### Implementation of hitting times of discrete time quantum random walks on Cubelike graphs [article]

Jaideep Mulherkar, Rishikant Rajdeepak, V Sunitha
<span title="2021-08-31">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We verify the results about the one-shot hitting time of quantum walks on a hypercube as proved in [https://link.springer.com/article/10.1007/s00440-004-0423-2].  ...  That is, for any cubelike graph of degree Δ, the probability of finding the quantum random walk at the target node at time πΔ/2 approaches 1 as the degree Δ of the cubelike graph approaches infinity.  ...  circuits to find the hitting times in cubelike graphs.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2108.13769v1">arXiv:2108.13769v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/im4opkvhyzcivlfqa27jbk5dpq">fatcat:im4opkvhyzcivlfqa27jbk5dpq</a> </span>
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### On the universal Gröbner bases of toric ideals of graphs

Christos Tatakis, Apostolos Thoma
<span title="">2011</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/z77xaqun7bcxjkh75wb7iseaty" style="color: black;">Journal of combinatorial theory. Series A</a> </i> &nbsp;
We characterize in graph theoretical terms the elements of the universal Gröbner basis of the toric ideal of a graph. We also provide a new degree bound.  ...  Finally, we give examples of graphs for which the true degrees of their circuits are less than the degrees of some elements of the Graver basis.  ...  The simplest example of a walk w such that B w is in the Graver basis but not in the universal Gröbner basis is the one with degree 6 whose graph is in Fig. 2 .  ...
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### Robust Graph Ideals [article]

Adam Boocher, Bryan Christopher Brown, Timothy Duff, Laura Lyman, Takumi Murayama, Amy Nesky, Karl Schaefer
<span title="2013-09-29">2013</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Our characterization shows that robustness can be determined solely in terms of graph-theoretic conditions on the set of circuits of G.  ...  We show that any robust toric ideal arising from a graph G is also minimally generated by its Graver basis. We then completely characterize all graphs which give rise to robust ideals.  ...  The following theorem is stated in terms of graph theoretic properties of the circuits of the graph G.  ...
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