scholarly journals Time and space efficient quantum algorithms for detecting cycles and testing bipartiteness

2018 ◽  
Vol 18 (1&2) ◽  
pp. 18-50 ◽  
Author(s):  
Chris Cade ◽  
Ashley Montanaro ◽  
Aleksandrs Belovs
Keyword(s):  
Matrix Model ◽  
Adjacency Matrix ◽  
Maximum Degree ◽  
Quantum Algorithms ◽  
Input Graph ◽  
Time And Space ◽  
Quantum Bits ◽  
Graph Problems ◽  
Vertex Graph ◽  
Connectivity Problem

We study space and time-efficient quantum algorithms for two graph problems – deciding whether an n-vertex graph is a forest, and whether it is bipartite. Via a reduction to the s-t connectivity problem, we describe quantum algorithms for deciding both properties in O˜(n 3/2 ) time whilst using O(log n) classical and quantum bits of storage in the adjacency matrix model. We then present quantum algorithms for deciding the two properties in the adjacency array model, which run in time O˜(n √ dm) and also require O(log n) space, where dm is the maximum degree of any vertex in the input graph.

Irredundance and Maximum Degree in Graphs

1997 ◽  
Vol 6 (2) ◽  
pp. 153-157 ◽  
Author(s):  
E. J. COCKAYNE ◽  
C. M. MYNHARDT
Keyword(s):  
Maximum Degree ◽  
Vertex Graph

It is proved that the smallest cardinality among the maximal irredundant sets in an n–vertex graph with maximum degree Δ([ges ]2) is at least 2n/3Δ. This substantially improves a bound by Bollobás and Cockayne [1]. The class of graphs which attain this bound is characterised.

scholarly journals Fast Algorithms for Diameter-Optimally Augmenting Paths and Trees

2019 ◽  
Vol 30 (02) ◽  
pp. 293-313
Author(s):  
Ulrike Große ◽  
Christian Knauer ◽  
Fabian Stehn ◽  
Joachim Gudmundsson ◽  
Michiel Smid
Keyword(s):  
Metric Space ◽  
Fast Algorithms ◽  
Exact Algorithms ◽  
Input Graph ◽  
Additional Edge ◽  
Vertex Graph

We consider the problem of augmenting an [Formula: see text]-vertex graph embedded in a metric space, by inserting one additional edge in order to minimize the diameter of the resulting graph. We present exact algorithms for the cases when (i) the input graph is a path, running in [Formula: see text] time, and (ii) the input graph is a tree, running in [Formula: see text] time. We also present an algorithm for paths that computes a [Formula: see text]-approximation in [Formula: see text] time.

scholarly journals Optimal induced universal graphs for bounded-degree graphs

2017 ◽  
Vol 166 (1) ◽  
pp. 61-74 ◽  
Author(s):  
NOGA ALON ◽  
RAJKO NENADOV
Keyword(s):  
Maximum Degree ◽  
Constant Factor ◽  
Induced Subgraph ◽  
Bounded Degree ◽  
Vertex Graph ◽  
Bounded Degree Graphs ◽  
Universal Graphs

AbstractWe show that for any constant Δ ≥ 2, there exists a graph Γ withO(nΔ / 2) vertices which contains everyn-vertex graph with maximum degree Δ as an induced subgraph. For odd Δ this significantly improves the best-known earlier bound and is optimal up to a constant factor, as it is known that any such graph must have at least Ω(nΔ/2) vertices.

Trees with given maximum degree minimizing the spectral radius

2016 ◽  
Vol 31 ◽  
pp. 335-361
Author(s):  
Xue Du ◽  
Lingsheng Shi
Keyword(s):  
Spectral Radius ◽  
Adjacency Matrix ◽  
Maximum Degree ◽  
Equal Length ◽  
Disjoint Paths ◽  
Common Vertex ◽  
Largest Eigenvalue ◽  
Large N ◽  
The Largest Eigenvalue

The spectral radius of a graph is the largest eigenvalue of the adjacency matrix of the graph. Let $T^*(n,\Delta ,l)$ be the tree which minimizes the spectral radius of all trees of order $n$ with exactly $l$ vertices of maximum degree $\Delta $. In this paper, $T^*(n,\Delta ,l)$ is determined for $\Delta =3$, and for $l\le 3$ and $n$ large enough. It is proven that for sufficiently large $n$, $T^*(n,3,l)$ is a caterpillar with (almost) uniformly distributed legs, $T^*(n,\Delta ,2)$ is a dumbbell, and $T^*(n,\Delta ,3)$ is a tree consisting of three distinct stars of order $\Delta $ connected by three disjoint paths of (almost) equal length from their centers to a common vertex. The unique tree with the largest spectral radius among all such trees is also determined. These extend earlier results of Lov\' asz and Pelik\'an, Simi\' c and To\u si\' c, Wu, Yuan and Xiao, and Xu, Lin and Shu.

scholarly journals A spanning bandwidth theorem in random graphs

2021 ◽  
pp. 1-31
Author(s):  
Peter Allen ◽  
Julia Böttcher ◽  
Julia Ehrenmüller ◽  
Jakob Schnitzer ◽  
Anusch Taraz
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Spanning Subgraph ◽  
Vertex Graph ◽  
Special Case

Abstract The bandwidth theorem of Böttcher, Schacht and Taraz states that any n-vertex graph G with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)n$ contains all n-vertex k-colourable graphs H with bounded maximum degree and bandwidth o(n). Recently, a subset of the authors proved a random graph analogue of this statement: for $p\gg \big(\tfrac{\log n}{n}\big)^{1/\Delta}$ a.a.s. each spanning subgraph G of G(n,p) with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)pn$ contains all n-vertex k-colourable graphs H with maximum degree $\Delta$ , bandwidth o(n), and at least $C p^{-2}$ vertices not contained in any triangle. This restriction on vertices in triangles is necessary, but limiting. In this paper, we consider how it can be avoided. A special case of our main result is that, under the same conditions, if additionally all vertex neighbourhoods in G contain many copies of $K_\Delta$ then we can drop the restriction on H that $Cp^{-2}$ vertices should not be in triangles.

scholarly journals The Spectral Radius and the Maximum Degree of Irregular Graphs

10.37236/956 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Sebastian M. Cioabă
Keyword(s):  
Spectral Radius ◽  
Regular Graph ◽  
Adjacency Matrix ◽  
Maximum Degree ◽  
Largest Eigenvalue ◽  
Irregular Graphs ◽  
The Largest Eigenvalue

Let $G$ be an irregular graph on $n$ vertices with maximum degree $\Delta$ and diameter $D$. We show that $$ \Delta-\lambda_1>{1\over nD}, $$ where $\lambda_1$ is the largest eigenvalue of the adjacency matrix of $G$. We also study the effect of adding or removing few edges on the spectral radius of a regular graph.

scholarly journals Expanders Are Universal for the Class of All Spanning Trees

2013 ◽  
Vol 22 (2) ◽  
pp. 253-281 ◽  
Author(s):  
DANIEL JOHANNSEN ◽  
MICHAEL KRIVELEVICH ◽  
WOJCIECH SAMOTIJ
Keyword(s):  
Random Graphs ◽  
Maximum Degree ◽  
Spanning Trees ◽  
Clique Number ◽  
Interesting Consequence ◽  
Random Regular Graph ◽  
Vertex Graph ◽  
Graph Class ◽  
Universal Graphs ◽  
Natural Expansion

A graph is calleduniversalfor a given graph class(or, equivalently,-universal) if it contains a copy of every graph inas a subgraph. The construction of sparse universal graphs for various classeshas received a considerable amount of attention. There is particular interest in tight-universal graphs, that is, graphs whose number of vertices is equal to the largest number of vertices in a graph from. Arguably, the most studied case is that whenis some class of trees. In this work, we are interested in(n,Δ), the class of alln-vertex trees with maximum degree at most Δ. We show that everyn-vertex graph satisfying certain natural expansion properties is(n,Δ)-universal. Our methods also apply to the case when Δ is some function ofn. Since random graphs are known to be good expanders, our result implies, in particular, that there exists a positive constantcsuch that the random graphG(n,cn−1/3log2n) is asymptotically almost surely (a.a.s.) universal for(n,O(1)). Moreover, a corresponding result holds for the random regular graph of degreecn2/3log2n. Another interesting consequence is the existence of locally sparsen-vertex(n,Δ)-universal graphs. For example, we show that one can (randomly) constructn-vertex(n,O(1))-universal graphs with clique number at most five. This complements the construction of Bhatt, Chung, Leighton and Rosenberg (1989), whose(n,Δ)-universal graphs with merelyO(n)edges contain large cliques of size Ω(Δ). Finally, we show that random graphs are robustly(n,Δ)-universal in the context of the Maker–Breaker tree-universality game.

Graph Colouring with No Large Monochromatic Components

2008 ◽  
Vol 17 (4) ◽  
pp. 577-589 ◽  
Author(s):  
NATHAN LINIAL ◽  
JIŘÍ MATOUŠEK ◽  
OR SHEFFET ◽  
GÁBOR TARDOS
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
The Other ◽  
Graph Colouring ◽  
Connected Subgraph ◽  
Regular Graphs ◽  
Maximum Order ◽  
Asymptotically Optimal ◽  
Vertex Graph ◽  
Order Of Magnitude

For a graph G and an integer t we let mcct(G) be the smallest m such that there exists a colouring of the vertices of G by t colours with no monochromatic connected subgraph having more than m vertices. Let be any non-trivial minor-closed family of graphs. We show that mcc2(G) = O(n2/3) for any n-vertex graph G ∈ . This bound is asymptotically optimal and it is attained for planar graphs. More generally, for every such , and every fixed t we show that mcct(G)=O(n2/(t+1)). On the other hand, we have examples of graphs G with no Kt+3 minor and with mcct(G)=Ω(n2/(2t−1)).It is also interesting to consider graphs of bounded degrees. Haxell, Szabó and Tardos proved mcc2(G) ≤ 20000 for every graph G of maximum degree 5. We show that there are n-vertex 7-regular graphs G with mcc2(G)=Ω(n), and more sharply, for every ϵ > 0 there exists cϵ > 0 and n-vertex graphs of maximum degree 7, average degree at most 6 + ϵ for all subgraphs, and with mcc2(G) ≥ cϵn. For 6-regular graphs it is known only that the maximum order of magnitude of mcc2 is between $\sqrt n$ and n.We also offer a Ramsey-theoretic perspective of the quantity mcct(G).

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