The Network Analysis of Urban Streets: A Primal Approach

The network metaphor in the analysis of urban and territorial cases has a long tradition especially in transportation/land-use planning and economic geography. More recently, urban design has brought its contribution by means of the "space syntax" methodology. All these approaches, though under different terms like accessibility, proximity, integration,connectivity, cost or effort, focus on the idea that some places (or streets) are more important than others because they are more central. The study of centrality in complex systems,however, originated in other scientific areas, namely in structural sociology, well before its use in urban studies; moreover, as a structural property of the system, centrality has never been extensively investigated metrically in geographic networks as it has been topologically in a wide range of other relational networks like social, biological or technological. After two previous works on some structural properties of the dual and primal graph representations of urban street networks (Porta et al. cond-mat/0411241; Crucitti et al. physics/0504163), in this paper we provide an in-depth investigation of centrality in the primal approach as compared to the dual one, with a special focus on potentials for urban design.Comment: 19 page, 4 figures. Paper related to the paper "The Network Analysis of Urban Streets: A Dual Approach" cond-mat/041124

Labelings of cubic graphs

V této bakalářské práci je zpracován přehled známých výsledků vybraných ohodnocení pro kubické grafy. Zaměřujeme se na tři ohodnocení magického typu, a to na supermagické ohodnocení, hranově magické totální ohodnocení a na vrcholově magické totální ohodnocení. Uvádíme specifické vlastnosti daných ohodnocení včetně souvisejících výpočtů a důkazů základních vlastností. Pro každé ohodnocení shrnujeme známé výsledky pro kubické grafy, spolu s konstrukcemi a důvody, proč u některých grafů ohodnocení nelze sestrojit. Dále zkoumáme souvislost mezi hranovým barvením kubických grafů a jejich ohodnoceními. V průběhu práce také pozorujeme, zda-li má planárnost grafu vliv na existenci, popř. neexistenci ohodnocení. Výsledkem je také výčet kubických grafů, resp. tříd kubických grafů, pro které výsledky zkoumaných ohodnocení nejsou známy.In this bachelor thesis the known results of selected magic graph labelings for cubic graphs are compiled. We focus our search to Supermagic labeling, Edge magic total labeling and Vertex magic total labeling. We demonstrate specific properties of named labelings by including basic calculations and proofs of known properties. For each labeling we present summary of known results for cubic graphs along with constructions of labelings for those graphs that are magic and proofs for those graphs, that do not have these magic labelings. We also look for relations between edge coloring of cubic graphs and existence of the labeling. In the course of this work we observe, if the planarity of a graph relates to the existence (or non-existence) of the labeling. As a result we present an overview of cubic graphs for which the existence or non-existence of the selected labelings is not known.470 - Katedra aplikované matematikyvýborn

Non-intersection of transient branching random walks

Funder: University of CambridgeAbstract: Let G be a Cayley graph of a nonamenable group with spectral radius ρ<1. It is known that branching random walk on G with offspring distribution μ is transient, i.e., visits the origin at most finitely often almost surely, if and only if the expected number of offspring μ¯ satisfies μ¯≤ρ-1. Benjamini and Müller (Groups Geom Dyn, 6:231–247, 2012) conjectured that throughout the transient supercritical phase 1<μ¯≤ρ-1, and in particular at the recurrence threshold μ¯=ρ-1, the trace of the branching random walk is tree-like in the sense that it is infinitely-ended almost surely on the event that the walk survives forever. This is essentially equivalent to the assertion that two independent copies of the branching random walk intersect at most finitely often almost surely. We prove this conjecture, along with several other related conjectures made by the same authors. A central contribution of this work is the introduction of the notion of local unimodularity, which we expect to have several further applications in the future

Non-intersection of transient branching random walks

Funder: University of CambridgeAbstract: Let G be a Cayley graph of a nonamenable group with spectral radius ρ<1. It is known that branching random walk on G with offspring distribution μ is transient, i.e., visits the origin at most finitely often almost surely, if and only if the expected number of offspring μ¯ satisfies μ¯≤ρ-1. Benjamini and Müller (Groups Geom Dyn, 6:231–247, 2012) conjectured that throughout the transient supercritical phase 1<μ¯≤ρ-1, and in particular at the recurrence threshold μ¯=ρ-1, the trace of the branching random walk is tree-like in the sense that it is infinitely-ended almost surely on the event that the walk survives forever. This is essentially equivalent to the assertion that two independent copies of the branching random walk intersect at most finitely often almost surely. We prove this conjecture, along with several other related conjectures made by the same authors. A central contribution of this work is the introduction of the notion of local unimodularity, which we expect to have several further applications in the future

Some results involving the AαA_\alpha-eigenvalues for graphs and line graphs

Let $G$ be a simple graph with adjacency matrix $A(G)$, signless Laplacian matrix $Q(G)$, degree diagonal matrix $D(G)$ and let $l(G)$ be the line graph of $G$. In 2017, Nikiforov defined the $A_\alpha$-matrix of $G$, $A_\alpha(G)$, as a linear convex combination of $A(G)$ and $D(G)$, the following way, $A_\alpha(G):=\alpha A(G)+(1-\alpha)D(G),$ where $\alpha\in[0,1]$. In this paper, we present some bounds for the eigenvalues of $A_\alpha(G)$ and for the largest and smallest eigenvalues of $A_\alpha(l(G))$. Extremal graphs attaining some of these bounds are characterized.Comment: 18 pages, 5 figures, 3 table

Detecting quantum speedup of random walks with machine learning

We explore the use of machine-learning techniques to detect quantum speedup in random walks on graphs. Specifically, we investigate the performance of three different neural-network architectures (variations on fully connected and convolutional neural networks) for identifying linear, cyclic, and random graphs that yield quantum speedups in terms of the hitting time for reaching a target node after starting in another node of the graph. Our results indicate that carefully building the data set for training can improve the performance of the neural networks, but all architectures we test struggle to classify large random graphs and generalize from training on one graph size to testing on another. If classification accuracy can be improved further, valuable insights about quantum advantage may be gleaned from these neural networks, not only for random walks, but more generally for quantum computing and quantum transport.Comment: 15 pages, 8 figure

On a game of chance in Marc Elsberg’s thriller “GREED”

A (possibly illegal) game of chance, which is described in Chap. 14 of Marc Elsberg’s thriller “GREED”, seems to offer an excellent chance of winning. However, as the gambling starts and evolves over several rounds, the actual experience of the vast majority of the gamblers in a pub is strikingly different. We provide an analysis of this specific game and several of its variants by elementary tools of probability. Thus we also encounter an interesting threshold phenomenon, which is related to the transition from a profit zone to a loss area. Our arguments are motivated and illustrated by numerical calculations with Python

A Creative Review on Coprime (Prime) Graphs

Coprime labelings and Coprime graphs have been of interest since 1980s and got popularized by the Entringer-Tout Tree Conjecture. Around the same time Newman&apos;s coprime mapping conjecture was settled by Pomerance and Selfridge. This result was further extended to integers in arithmetic progression. Since then coprime graphs were studied for various combinatorial properties. Here, coprimality of graphs for classes of graphs under the themes: Bipartite with special attention to Acyclicity, Eulerian and Regularity. Extremal graphs under non-coprimality and Eulerian properties are studied. Embeddings of coprime graphs in the general graphs, the maximum coprime graph and the Eulerian coprime graphs are studied as subgraphs and induced subgraphs. The purpose of this review is to assimilate the available works on coprime graphs. The results in the context of these themes are reviewed including embeddings and extremal problems