scholarly journals Intersections of Randomly Embedded Sparse Graphs are Poisson

10.37236/1468 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
Edward A. Bender ◽  
E. Rodney Canfield
Keyword(s):  
Maximum Degree ◽  
Spanning Trees ◽  
Sufficient Conditions ◽  
Intersection Theorem ◽  
Bounded Degree ◽  
Sparse Graphs ◽  
Inversion Theorem ◽  
Number Of Spanning Trees

Suppose that $t \ge 2$ is an integer, and randomly label $t$ graphs with the integers $1 \dots n$. We give sufficient conditions for the number of edges common to all $t$ of the labelings to be asymptotically Poisson as $n \to \infty$. We show by example that our theorem is, in a sense, best possible. For $G_n$ a sequence of graphs of bounded degree, each having at most $n$ vertices, Tomescu has shown that the number of spanning trees of $K_n$ having $k$ edges in common with $G_n$ is asymptotically $e^{-2s/n}(2s/n)^k/k! \times n^{n-2}$, where $s=s(n)$ is the number of edges in $G_n$. As an application of our Poisson-intersection theorem, we extend this result to the case in which maximum degree is only restricted to be ${\scriptstyle\cal O}(n \log\log n/\log n)$. We give an inversion theorem for falling moments, which we use to prove our Poisson-intersection theorem.

scholarly journals Spanning Trees of Bounded Degree

10.37236/1577 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Andrzej Czygrinow ◽  
Genghua Fan ◽  
Glenn Hurlbert ◽  
H. A. Kierstead ◽  
William T. Trotter
Keyword(s):  
Spanning Tree ◽  
Maximum Degree ◽  
Additional Assumption ◽  
Spanning Trees ◽  
Independent Set ◽  
Hamilton Cycle ◽  
Bounded Degree ◽  
Hamilton Path ◽  
Classic Theorem

Dirac's classic theorem asserts that if ${\bf G}$ is a graph on $n$ vertices, and $\delta({\bf G})\ge n/2$, then ${\bf G}$ has a hamilton cycle. As is well known, the proof also shows that if $\deg(x)+\deg(y)\ge(n-1)$, for every pair $x$, $y$ of independent vertices in ${\bf G}$, then ${\bf G}$ has a hamilton path. More generally, S. Win has shown that if $k\ge 2$, ${\bf G}$ is connected and $\sum_{x\in I}\deg(x)\ge n-1$ whenever $I$ is a $k$-element independent set, then ${\bf G}$ has a spanning tree ${\bf T}$ with $\Delta({\bf T})\le k$. Here we are interested in the structure of spanning trees under the additional assumption that ${\bf G}$ does not have a spanning tree with maximum degree less than $k$. We show that apart from a single exceptional class of graphs, if $\sum_{x\in I}\deg(x)\ge n-1$ for every $k$-element independent set, then ${\bf G}$ has a spanning caterpillar ${\bf T}$ with maximum degree $k$. Furthermore, given a maximum path $P$ in ${\bf G}$, we may require that $P$ is the spine of ${\bf T}$ and that the set of all vertices whose degree in ${\bf T}$ is $3$ or larger is independent in ${\bf T}$.

scholarly journals Partitioning Sparse Graphs into an Independent Set and a Forest of Bounded Degree

10.37236/6815 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
François Dross ◽  
Mickael Montassier ◽  
Alexandre Pinlou
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Independent Set ◽  
Average Degree ◽  
Bounded Degree ◽  
Sparse Graphs ◽  
Maximum Average Degree

An $({\cal I},{\cal F}_d)$-partition of a graph is a partition of the vertices of the graph into two sets $I$ and $F$, such that $I$ is an independent set and $F$ induces a forest of maximum degree at most $d$. We show that for all $M<3$ and $d \ge \frac{2}{3-M} - 2$, if a graph has maximum average degree less than $M$, then it has an $({\cal I},{\cal F}_d)$-partition. Additionally, we prove that for all $\frac{8}{3} \le M < 3$ and $d \ge \frac{1}{3-M}$, if a graph has maximum average degree less than $M$ then it has an $({\cal I},{\cal F}_d)$-partition. It follows that planar graphs with girth at least $7$ (resp. $8$, $10$) admit an $({\cal I},{\cal F}_5)$-partition (resp. $({\cal I},{\cal F}_3)$-partition, $({\cal I},{\cal F}_2)$-partition).

scholarly journals The average number of spanning trees in sparse graphs with given degrees

2017 ◽  
Vol 63 ◽  
pp. 6-25 ◽  
Author(s):  
Catherine Greenhill ◽  
Mikhail Isaev ◽  
Matthew Kwan ◽  
Brendan D. McKay
Keyword(s):  
Spanning Trees ◽  
Sparse Graphs ◽  
Number Of Spanning Trees

On the Kirchhoff Index of Graphs

2013 ◽  
Vol 68 (8-9) ◽  
pp. 531-538 ◽  
Author(s):  
Kinkar C. Das
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Spanning Trees ◽  
Upper Bounds ◽  
Type Result ◽  
Lower And Upper Bounds ◽  
Kirchhoff Index ◽  
Laplacian Eigenvalues ◽  
Number Of Spanning Trees

Let G be a connected graph of order n with Laplacian eigenvalues μ1 ≥ μ2 ≥ ... ≥ μn-1 > mn = 0. The Kirchhoff index of G is defined as [xxx] In this paper. we give lower and upper bounds on Kf of graphs in terms on n, number of edges, maximum degree, and number of spanning trees. Moreover, we present lower and upper bounds on the Nordhaus-Gaddum-type result for the Kirchhoff index.

Reliable Graphs for SLAM

2019 ◽  
Vol 38 (2-3) ◽  
pp. 260-298 ◽  
Author(s):  
Kasra Khosoussi ◽  
Matthew Giamou ◽  
Gaurav S Sukhatme ◽  
Shoudong Huang ◽  
Gamini Dissanayake ◽  
...  
Keyword(s):  
Spanning Trees ◽  
Graphical Representation ◽  
Graph Connectivity ◽  
Maximum Likelihood Estimates ◽  
Sparse Graphs ◽  
Localization And Mapping ◽  
Weighted Number ◽  
Graph Slam ◽  
The Impact ◽  
Number Of Spanning Trees

Estimation-over-graphs (EoG) is a class of estimation problems that admit a natural graphical representation. Several key problems in robotics and sensor networks, including sensor network localization, synchronization over a group, and simultaneous localization and mapping (SLAM) fall into this category. We pursue two main goals in this work. First, we aim to characterize the impact of the graphical structure of SLAM and related problems on estimation reliability. We draw connections between several notions of graph connectivity and various properties of the underlying estimation problem. In particular, we establish results on the impact of the weighted number of spanning trees on the D-optimality criterion in 2D SLAM. These results enable agents to evaluate estimation reliability based only on the graphical representation of the EoG problem. We then use our findings and study the problem of designing sparse SLAM problems that lead to reliable maximum likelihood estimates through the synthesis of sparse graphs with the maximum weighted tree connectivity. Characterizing graphs with the maximum number of spanning trees is an open problem in general. To tackle this problem, we establish several new theoretical results, including the monotone log-submodularity of the weighted number of spanning trees. We exploit these structures and design a complementary greedy–convex pair of efficient approximation algorithms with provable guarantees. The proposed synthesis framework is applied to various forms of the measurement selection problem in resource-constrained SLAM. Our algorithms and theoretical findings are validated using random graphs, existing and new synthetic SLAM benchmarks, and publicly available real pose-graph SLAM datasets.

scholarly journals Sharp upper bounds for the number of spanning trees of a graph

2008 ◽  
Vol 2 (2) ◽  
pp. 255-259 ◽  
Author(s):  
Lihua Feng ◽  
Yu Guihai ◽  
Zhengtao Jiang ◽  
Lingzhi Ren
Keyword(s):  
Maximum Degree ◽  
Spanning Trees ◽  
Upper Bounds ◽  
Number Of Spanning Trees

This note presents two new upper bounds for the number of spanning trees of a graph in terms of the order, edge number and maximum degree of a graph.

scholarly journals Clustered 3-colouring graphs of bounded degree

2021 ◽  
pp. 1-13
Author(s):  
Vida Dujmović ◽  
Louis Esperet ◽  
Pat Morin ◽  
Bartosz Walczak ◽  
David R. Wood
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Bounded Degree ◽  
Bounded Number ◽  
Proper Vertex ◽  
Vertex Colouring

Abstract A (not necessarily proper) vertex colouring of a graph has clustering c if every monochromatic component has at most c vertices. We prove that planar graphs with maximum degree $\Delta$ are 3-colourable with clustering $O(\Delta^2)$ . The previous best bound was $O(\Delta^{37})$ . This result for planar graphs generalises to graphs that can be drawn on a surface of bounded Euler genus with a bounded number of crossings per edge. We then prove that graphs with maximum degree $\Delta$ that exclude a fixed minor are 3-colourable with clustering $O(\Delta^5)$ . The best previous bound for this result was exponential in $\Delta$ .

scholarly journals The Complexity of Approximating the Matching Polynomial in the Complex Plane

2021 ◽  
Vol 13 (2) ◽  
pp. 1-37
Author(s):  
Ivona Bezáková ◽  
Andreas Galanis ◽  
Leslie Ann Goldberg ◽  
Daniel Štefankovič
Keyword(s):  
Real Axis ◽  
Complex Plane ◽  
Maximum Degree ◽  
Negative Real Axis ◽  
Positive Real ◽  
Bounded Degree ◽  
Matching Polynomial ◽  
Correlation Decay ◽  
Connective Constant ◽  
General Complex

We study the problem of approximating the value of the matching polynomial on graphs with edge parameter γ, where γ takes arbitrary values in the complex plane. When γ is a positive real, Jerrum and Sinclair showed that the problem admits an FPRAS on general graphs. For general complex values of γ, Patel and Regts, building on methods developed by Barvinok, showed that the problem admits an FPTAS on graphs of maximum degree Δ as long as γ is not a negative real number less than or equal to −1/(4(Δ −1)). Our first main result completes the picture for the approximability of the matching polynomial on bounded degree graphs. We show that for all Δ ≥ 3 and all real γ less than −1/(4(Δ −1)), the problem of approximating the value of the matching polynomial on graphs of maximum degree Δ with edge parameter γ is #P-hard. We then explore whether the maximum degree parameter can be replaced by the connective constant. Sinclair et al. showed that for positive real γ, it is possible to approximate the value of the matching polynomial using a correlation decay algorithm on graphs with bounded connective constant (and potentially unbounded maximum degree). We first show that this result does not extend in general in the complex plane; in particular, the problem is #P-hard on graphs with bounded connective constant for a dense set of γ values on the negative real axis. Nevertheless, we show that the result does extend for any complex value γ that does not lie on the negative real axis. Our analysis accounts for complex values of γ using geodesic distances in the complex plane in the metric defined by an appropriate density function.

scholarly journals Properties of Commuting Graphs over Semidihedral Groups

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 103
Author(s):  
Tao Cheng ◽  
Matthias Dehmer ◽  
Frank Emmert-Streib ◽  
Yongtao Li ◽  
Weijun Liu
Keyword(s):  
Spanning Trees ◽  
Laplacian Spectrum ◽  
Vertex Connectivity ◽  
Laplacian Energy ◽  
Number Of Spanning Trees

This paper considers commuting graphs over the semidihedral group SD8n. We compute their eigenvalues and obtain that these commuting graphs are not hyperenergetic for odd n≥15 or even n≥2. We further compute the Laplacian spectrum, the Laplacian energy and the number of spanning trees of the commuting graphs over SD8n. We also discuss vertex connectivity, planarity, and minimum disconnecting sets of these graphs and prove that these commuting graphs are not Hamiltonian.

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