Johnson graph in Grassmann graph

Two urns initially contain r red balls and n – r black balls respectively. At each time epoch a ball is chosen randomly from each urn and the balls are switched. Effectively the same process arises in many other contexts, notably for a symmetric exclusion process and random walk on the Johnson graph. If Y(·) counts the number of black balls in the first urn then we give a direct asymptotic analysis of its transition probabilities to show that (when run at rate (n – r)/n in continuous time) for as n →∞, where π n denotes the equilibrium distribution of Y(·) and γ α = 1 – α /β (1 – β). Thus for large n the transient probabilities approach their equilibrium values at time log n + log|γ α | (≦log n) in a particularly sharp manner. The same is true of the separation distance between the transient distribution and the equilibrium distribution. This is an explicit analysis of the so-called cut-off phenomenon associated with a wide variety of Markov chains.

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Perfect 2-Colorings of the Grassmann Graph of Planes

The Electronic Journal of Combinatorics ◽

10.37236/8672 ◽

2020 ◽

Vol 27 (1) ◽

Author(s):

Stefaan De Winter ◽

Klaus Metsch

Keyword(s):

Infinite Family ◽

Grassmann Graph ◽

The Family

We construct an infinite family of intriguing sets, or equivalently perfect 2-colorings, that are not tight in the Grassmann graph of planes of PG$(n,q)$, $n\ge 5$ odd, and show that the members of the family are the smallest possible examples if $n\ge 9$ or $q\ge 25$.

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On the connectedness and diameter of a geometric Johnson graph

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.613 ◽

2013 ◽

Vol Vol. 15 no. 3 (Combinatorics) ◽

Author(s):

Crevel Bautista-Santiago ◽

Javier Cano ◽

Ruy Fabila-Monroy ◽

David Flores-Peñaloza ◽

Hernàn González-Aguilar ◽

...

Keyword(s):

Lower Bounds ◽

General Position ◽

Convex Set ◽

Upper And Lower Bounds ◽

Johnson Graph ◽

International Audience

Combinatorics International audience Let P be a set of n points in general position in the plane. A subset I of P is called an island if there exists a convex set C such that I = P \C. In this paper we define the generalized island Johnson graph of P as the graph whose vertex consists of all islands of P of cardinality k, two of which are adjacent if their intersection consists of exactly l elements. We show that for large enough values of n, this graph is connected, and give upper and lower bounds on its diameter.

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Cyclic Decomposition of $k$-Permutations and Eigenvalues of the Arrangement Graphs

The Electronic Journal of Combinatorics ◽

10.37236/3711 ◽

2013 ◽

Vol 20 (4) ◽

Cited By ~ 10

Author(s):

Bai Fan Chen ◽

Ebrahim Ghorbani ◽

Kok Bin Wong

Keyword(s):

Adjacency Matrix ◽

Johnson Graph ◽

Equitable Partition ◽

Smallest Eigenvalue ◽

Arrangement Graph ◽

Complete Set ◽

Element Set

The $(n,k)$-arrangement graph $A(n,k)$ is a graph with all the $k$-permutations of an $n$-element set as vertices where two $k$-permutations are adjacent if they agree in exactly $k-1$ positions. We introduce a cyclic decomposition for $k$-permutations and show that this gives rise to a very fine equitable partition of $A(n,k)$. This equitable partition can be employed to compute the complete set of eigenvalues (of the adjacency matrix) of $A(n,k)$. Consequently, we determine the eigenvalues of $A(n,k)$ for small values of $k$. Finally, we show that any eigenvalue of the Johnson graph $J(n,k)$ is an eigenvalue of $A(n,k)$ and that $-k$ is the smallest eigenvalue of $A(n,k)$ with multiplicity ${\cal O}(n^k)$ for fixed $k$.

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Approach to Stationarity of the Bernoulli–Laplace Diffusion Model

Advances in Applied Probability ◽

10.1017/s0001867800026513 ◽

1994 ◽

Vol 26 (03) ◽

pp. 715-727

Author(s):

Peter Donnelly ◽

Peter Lloyd ◽

Aidan Sudbury

Keyword(s):

Continuous Time ◽

Equilibrium Distribution ◽

Transition Probabilities ◽

Exclusion Process ◽

Separation Distance ◽

Johnson Graph ◽

Transient Distribution ◽

Transient Probabilities ◽

Explicit Analysis ◽

Laplace Diffusion

Two urns initially contain r red balls and n – r black balls respectively. At each time epoch a ball is chosen randomly from each urn and the balls are switched. Effectively the same process arises in many other contexts, notably for a symmetric exclusion process and random walk on the Johnson graph. If Y(·) counts the number of black balls in the first urn then we give a direct asymptotic analysis of its transition probabilities to show that (when run at rate (n – r)/n in continuous time) for as n →∞, where π n denotes the equilibrium distribution of Y(·) and γ α = 1 – α /β (1 – β). Thus for large n the transient probabilities approach their equilibrium values at time log n + log|γ α | (≦log n) in a particularly sharp manner. The same is true of the separation distance between the transient distribution and the equilibrium distribution. This is an explicit analysis of the so-called cut-off phenomenon associated with a wide variety of Markov chains.

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SCALING LIMIT FOR GIBBS STATES OF JOHNSON GRAPHS AND RESULTING MEIXNER CLASSES

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025703001080 ◽

2003 ◽

Vol 06 (01) ◽

pp. 139-143 ◽

Cited By ~ 3

Author(s):

AKIHITO HORA

Keyword(s):

Fock Space ◽

Gibbs State ◽

Scaling Limit ◽

Temperature Limit ◽

Zero Temperature ◽

Johnson Graph ◽

Meixner Polynomials ◽

Johnson Graphs ◽

The One ◽

Interacting Fock Space

Asymptotic behavior of spectral distribution of the adjacency operator on the Johnson graph with respect to the Gibbs state is discussed in infinite volume and zero temperature limit. The limit picture is drawn on the one-mode interacting Fock space associated with Meixner polynomials.

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