Johnson Graphs Research Articles

Mutual-visibility problems in Kneser and Johnson graphs

Mutual-visibility problems in Kneser and Johnson graphs

New Karhunen-Loève expansions based on Hahn polynomials with application to a Sobolev test for uniformity on Johnson graphs

New Karhunen-Loève expansions based on Hahn polynomials with application to a Sobolev test for uniformity on Johnson graphs

Gromov hyperbolicity of Johnson and Kneser graphs

The concept of Gromov hyperbolicity is a geometric concept that leads to a rich general theory. Johnson and Kneser graphs are interesting combinatorial graphs defined from systems of sets. In this work we compute the precise value of the hyperbolicity constant of every Johnson graph. Also, we obtain good bounds on the hyperbolicity constant of every Kneser graph, and in many cases, we even compute its precise value.

Failed zero forcing numbers of Kneser graphs, Johnson graphs, and hypercubes

Failed zero forcing numbers of Kneser graphs, Johnson graphs, and hypercubes

The odd girth of generalized Johnson graphs

For any non-negative integers v>k>i, the generalized Johnson graph, X=J(v,k,i), is the graph whose vertices are the k-subsets of a v-set, and where any two vertices A and B are adjacent whenever |A∩B|=i. In this note, we prove that if v≥2k and (v,k,i)≠(2k,k,0), then the odd girth of X is given by:og(X)=2⌈k−iv−2k+2i⌉+1.

Estimates of the Number of Edges in Subgraphs of Johnson Graphs

Estimates of the Number of Edges in Subgraphs of Johnson Graphs

On the spectra and spectral radii of token graphs

Let G be a graph on n vertices. The k-token graph (or symmetric k-th power) of G, denoted by Fk(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(G)$$\\end{document}, has as vertices the nk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${n\\atopwithdelims ()k}$$\\end{document}k-subsets of vertices from G, and two vertices are adjacent when their symmetric difference is a pair of adjacent vertices in G. In particular, Fk(Kn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(K_n)$$\\end{document} is the Johnson graph J(n, k), which is a distance-regular graph used in coding theory. In this paper, we present some results concerning the (adjacency and Laplacian) spectrum of Fk(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(G)$$\\end{document} in terms of the spectrum of G. For instance, when G is walk-regular, an exact value for the spectral radius ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho $$\\end{document} (or maximum eigenvalue) of Fk(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(G)$$\\end{document} is obtained. When G is distance-regular, other eigenvalues of its 2-token graph are derived using the theory of equitable partitions. A generalization of Aldous’ spectral gap conjecture (which is now a theorem) is proposed.

Lackadaisical discrete-time quantum walk on Johnson graph

Quantum walk is widely used in search problems, and it has an ideal acceleration effect compared with classical algorithm. Suppose there is a marked vertex w on Johnson graph J(n,k), we hope to find w using quantum walk. We adopt the coined quantum walk model, and we have proved that, for fixed order k, lackadaisical discrete-time quantum walk (DTQW) can find w with success probability 1−o(1), keeping a quadratic speedup. Before this paper, the best result of DTQW is that discrete-time quantum walk finds w with success probability 12−o(1). We improve the success probability to 1−o(1) by adding self-loops with proper weights and choose the appropriate oracle operator. Our result also matches the result of continuous-time quantum walk.

Estimates of the Number of Edges in Subgraphs of Johnson Graphs

В работе мы рассматриваем специальные дистанционные графы и оцениваем число ребер в их подграфах. Полученные оценки улучшают известные результаты. Библиография: 19 названий.

Lower and upper bounds for the minimum number of edges in some subgraphs of the Johnson graph

Получены нижние и верхние оценки минимального числа ребер в индуцированных подграфах с $l$ вершинами графа $G(n,3,1)$, где $l \sim cn^2$. Полученные результаты улучшают ранее доказанные оценки этой величины в данном режиме. Библиография: 16 названий.

Finding the Fixing Number of Johnson Graphs J(n, k) for k Є {2; 3}

The graph invariant, aptly named the fixing number, is the smallest number of vertices that, when fixed, eliminate all non-trivial automorphisms (or symmetries) of a graph. Although many graphs have established fixing numbers, Johnson graphs, a family of graphs related to the graph isomorphism problem, have only partially classified fixing numbers. By examining specific orbit sizes of the automorphism group of Johnson graphs and classifying the subsequent remaining subgroups of the automorphism group after iteratively fixing vertices, we provide exact minimal sequences of fixed vertices, in turn establishing the fixing number of infinitely many Johnson graphs. KEYWORDS: Graph Automorphism Groups; Symmetry Breaking; Fixing Number; Determining Number; Johnson Graphs; Kneser Graphs; Graph Invariants; Permutation Groups; Minimal Sized Bases.

Neighbour-transitive codes in Kneser graphs

A code C is a subset of the vertex set of a graph and C is s-neighbour-transitive if its automorphism group Aut(C) acts transitively on each of the first s+1 parts C0,C1,…,Cs of the distance partition{C=C0,C1,…,Cρ}, where ρ is the covering radius of C. While codes have traditionally been studied in the Hamming and Johnson graphs, we consider here codes in the Kneser graphs. Let Ω be the underlying set on which the Kneser graph K(n,k) is defined. Our first main result says that if C is a 2-neighbour-transitive code in K(n,k) such that C has minimum distance at least 5, then n=2k+1 (i.e., C is a code in an odd graph) and C lies in a particular infinite family or is one particular sporadic example. We then prove several results when C is a neighbour-transitive code in the Kneser graph K(n,k). First, if Aut(C) acts intransitively on Ω we characterise C in terms of certain parameters. We then assume that Aut(C) acts transitively on Ω, first proving that if C has minimum distance at least 3 then either K(n,k) is an odd graph or Aut(C) has a 2-homogeneous (and hence primitive) action on Ω. We then assume that C is a code in an odd graph and Aut(C) acts imprimitively on Ω and characterise C in terms of certain parameters. We give examples in each of these cases and pose several open problems.

The Hamilton Compression of Highly Symmetric Graphs

We say that a Hamilton cycle C=(x1,…,xn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C=(x_1,\\ldots ,x_n)$$\\end{document} in a graph G is k-symmetric, if the mapping xi↦xi+n/k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_i\\mapsto x_{i+n/k}$$\\end{document} for all i=1,…,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$i=1,\\ldots ,n$$\\end{document}, where indices are considered modulo n, is an automorphism of G. In other words, if we lay out the vertices x1,…,xn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_1,\\ldots ,x_n$$\\end{document} equidistantly on a circle and draw the edges of G as straight lines, then the drawing of G has k-fold rotational symmetry, i.e., all information about the graph is compressed into a 360∘/k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$360^\\circ /k$$\\end{document} wedge of the drawing. The maximum k for which there exists a k-symmetric Hamilton cycle in G is referred to as the Hamilton compression of G. We investigate the Hamilton compression of four different families of vertex-transitive graphs, namely hypercubes, Johnson graphs, permutahedra and Cayley graphs of abelian groups. In several cases, we determine their Hamilton compression exactly, and in other cases, we provide close lower and upper bounds. The constructed cycles have a much higher compression than several classical Gray codes known from the literature. Our constructions also yield Gray codes for bitstrings, combinations and permutations that have few tracks and/or that are balanced.

The automorphism groups of some token graphs

The token graphs of graphs have been studied at least from the 80’s with different names and by different authors. The Johnson graph J(n, k) is isomorphic to the k-token graph of the complete graph Kn. To our knowledge, the unique results about the automorphism groups of token graphs are for the case of the Johnson graphs. In this paper we begin the study of the automorphism groups of token graphs of another graphs. In particular we obtain the automorphism group of the k-token graph of the path graph Pn, for n 6≠ 2k. Also, we obtain the automorphism group of the 2-token graph of the following graphs: cycle, star, fan and wheel graphs.

The Clebsch–Gordan coefficients of [formula omitted] and the Terwilliger algebras of Johnson graphs

The universal enveloping algebra U(sl2) of sl2 is a unital associative algebra over C generated by E,F,H subject to the relations[H,E]=2E,[H,F]=−2F,[E,F]=H. The elementΛ=EF+FE+H22 is called the Casimir element of U(sl2). Let Δ:U(sl2)→U(sl2)⊗U(sl2) denote the comultiplication of U(sl2). The universal Hahn algebra H is a unital associative algebra over C generated by A,B,C and the relations assert that [A,B]=C and each of[C,A]+2A2+B,[B,C]+4BA+2C is central in H. Inspired by the Clebsch–Gordan coefficients of U(sl2), we discover an algebra homomorphism ♮:H→U(sl2)⊗U(sl2) that mapsA↦H⊗1−1⊗H4,B↦Δ(Λ)2,C↦E⊗F−F⊗E. By pulling back via ♮ any U(sl2)⊗U(sl2)-module can be considered as an H-module. For any integer n≥0 there exists a unique (n+1)-dimensional irreducible U(sl2)-module Ln up to isomorphism. We study the decomposition of the H-module Lm⊗Ln for any integers m,n≥0. We link these results to the Terwilliger algebras of Johnson graphs. We express the dimensions of the Terwilliger algebras of Johnson graphs in terms of binomial coefficients.

On the Super (Edge)-Connectivity of Generalized Johnson Graphs

Let [Formula: see text], [Formula: see text] and [Formula: see text] be non-negative integers. The generalized Johnson graph [Formula: see text] is the graph whose vertices are the [Formula: see text]-subsets of the set [Formula: see text], and two vertices are adjacent if and only if they intersect with [Formula: see text] elements. Special cases of generalized Johnson graph include the Kneser graph [Formula: see text] and the Johnson graph [Formula: see text]. These graphs play an important role in coding theory, Ramsey theory, combinatorial geometry and hypergraphs theory. In this paper, we discuss the connectivity properties of the Kneser graph [Formula: see text] and [Formula: see text] by their symmetric properties. Specifically, with the help of some properties of vertex/edge-transitive graphs we prove that [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text] are super restricted edge-connected. Besides, we obtain the exact value of the restricted edge-connectivity and the cyclic edge-connectivity of the Kneser graph [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text], and further show that the Kneser graph [Formula: see text] [Formula: see text] is super vertex-connected and super cyclically edge-connected.

On distance-regular graphs Γ of diameter 3 for which Γ 3 is a triangle-free graph

Abstract There exist well-known distance-regular graphs Γ of diameter 3 for which Γ 3 is a triangle-free graph. An example is given by the Johnson graph J (8, 3) with the intersection array {15, 8, 3;1, 4, 9}. The paper is concerned with the problem of the existence of distance-regular graphs Γ with the intersection arrays {78, 50, 9;1, 15, 60} and {174, 110, 18;1, 30, 132} for which Γ 3 is a triangle-free graph.

Large cycles in generalized Johnson graphs

AbstractWe count cycles of an unbounded length in generalized Johnson graphs. Asymptotics of the number of such cycles is obtained for certain growth rates of the cycle length.

Aplikasi SABARSON (Si Penjaga Kualitas Rumah Sakit) Sebagai Early Warning System

Determining the quality of hospital service is about the efficiency of bed management. In some hospitals, Barber Johnson graphs are still described in Microsoft Excel, but the BOR indicator has not been displayed. SABARSON: The Guardian of Hospital Quality as an Early Warning System is a creative innovation in system design that can be used to determine the quality of a hospital by displaying a graph of the four indicators BOR, LOS, BTO, and TOI. This type of design uses the system development method with the development life cycle (SDLC) approach with the stage of problem identification, reference search, dataflow and concept preparation, color palette preparation, system design, and system implementation. The purpose of this study is to help and facilitate medical record workers in describing or displaying a Barber Johnson graph quickly and precisely. This study resulted in several interface design results such as login, BOR, TOI, LOS, BTO, report recapitulation, and Barber Johnson graphic.

Entanglement of free fermions on Johnson graphs

Free fermions on Johnson graphs J(n, k) are considered, and the entanglement entropy of sets of neighborhoods is computed. For a subsystem composed of a single neighborhood, an analytical expression is provided by the decomposition in irreducible submodules of the Terwilliger algebra of J(n, k) embedded in two copies of su(2). For a subsystem composed of multiple neighborhoods, the construction of a block-tridiagonal operator that commutes with the entanglement Hamiltonian is presented, its usefulness in computing the entropy is stressed, and the area law pre-factor is discussed.