Johnson Graphs Research Articles

Notes on simplicial rook graphs

The simplicial rook graph $$\mathrm{SR}(m,n)$$SR(m,n) is the graph of which the vertices are the sequences of nonnegative integers of length m summing to n, where two such sequences are adjacent when they differ in precisely two places. We show that $$\mathrm{SR}(m,n)$$SR(m,n) has integral eigenvalues, and smallest eigenvalue $$s = \max \left( -n, -{m \atopwithdelims ()2}\right) $$s=max-n,-m2, and that this graph has a large part of its spectrum in common with the Johnson graph $$J(m+n-1,n)$$J(m+n-1,n). We determine the automorphism group and several other properties.

On the ensemble of optimal identifying codes in a twin-free graph

Let G = (V, E) be a graph. For v ? V and r ? 1, we denote by BG,r(v) the ball of radius r and centre v. A set C⊆V$C \subseteq V$ is said to be an r-identifying code if the sets BG,r(v)?C$B_{G,r}(v)\cap C$, v ? V, are all nonempty and distinct. A graph G which admits an r-identifying code is called r-twin-free, and in this case the smallest size of an r-identifying code is denoted by ?r(G). We study the ensemble of all the different optimalr-identifying codes C, i.e., such that |C| = ?r(G). We show that, given any collection A$\mathcal {A}$ of k-subsets of V1={1,2,?,n}, there is a positive integer m, a graph G = (V, E) with V=V1?V2$V=V_{1}\cup V_{2}$, where V2 = {n + 1 ,?, n + m}, and a set S⊆V2$S\subseteq V_{2}$ such that C⊆V$C\subseteq V$ is an optimal r-identifying code in G if, and only if, C=A?S$C=A\cup S$ for some A?A$A\in \mathcal {A}$. This result gives a direct connection with induced subgraphs of Johnson graphs, which are graphs with vertex set a collection of k-subsets of V1, with edges between any two vertices sharing k?1 elements.

The Terwilliger algebra of the incidence graphs of Johnson geometry, II

Let J(n,m,m+1) denote the incidence graph of Johnson geometry. It is well known that the Johnson graph J(n,m) is a halved graph of J(n,m,m+1). Let T and T′ be the Terwilliger algebra of J(n,m,m+1) and J(n,m), respectively. In this paper, we focus on the structures of irreducible T-modules, and then completely determine T. Furthermore, we discover the relationship between irreducible modules of T and T′. As a result, the algebra T′ is determined again.

On the Number of Matroids Compared to the Number of Sparse Paving Matroids

It has been conjectured that sparse paving matroids will eventually predominate in any asymptotic enumeration of matroids, i.e. that $\lim_{n\rightarrow\infty} s_n/m_n = 1$, where $m_n$ denotes the number of matroids on $n$ elements, and $s_n$ the number of sparse paving matroids. In this paper, we show that $$\lim_{n\rightarrow \infty}\frac{\log s_n}{\log m_n}=1.$$ We prove this by arguing that each matroid on $n$ elements has a faithful description consisting of a stable set of a Johnson graph together with a (by comparison) vanishing amount of other information, and using that stable sets in these Johnson graphs correspond one-to-one to sparse paving matroids on $n$ elements.As a consequence of our result, we find that for all $\beta > \displaystyle{\sqrt{\frac{\ln 2}{2}}} = 0.5887\cdots$, asymptotically almost all matroids on $n$ elements have rank in the range $n/2 \pm \beta\sqrt{n}$.

A generalisation of Johnson graphs with an application to triple factorisations

In this paper, we introduce a new generalisation of Johnson graphs. The study of these graphs is linked to the study of intransitive triple factorisations Sym(Ω)=ABA of the (finite) symmetric group, where the subgroups A and B are intransitive subgroups of Sym(Ω). Indeed, we give combinatorial arguments to investigate the conditions under which such factorisations exist. We also use combinatorial arguments to study those conditions for which Sym(Ω) is a Geometric ABA-group, that is to say, Sym(Ω)=ABA, A⊈B, B⊈A and AB∩BA=A∪B.

On the distance spectrum of distance regular graphs

The distance matrix of a simple graph G is D(G)=(dij), where dij is the distance between ith and jth vertices of G. The spectrum of the distance matrix is known as the distance spectrum or D-spectrum of G. A simple connected graph G is called distance regular if it is regular, and if for any two vertices x,y∈G at distance i, there are constant number of neighbors ci and bi of y at distance i−1 and i+1 from x respectively. In this paper we prove that distance regular graphs with diameter d have at most d+1 distinct D-eigenvalues. We find an equitable partition and the corresponding quotient matrix of the distance regular graph which gives the distinct D-eigenvalues. We also prove that distance regular graphs satisfying bi=cd−1 have at most ⌈d2⌉+2 distinct D-eigenvalues. Applying these results we find the distance spectrum of some distance regular graphs including the well known Johnson graphs. Finally we also answer the questions asked by Lin et al. [16].

Groups that are transitive on all partitions of a given shape

Let $$[n] = K_1\dot{\cup }K_2 \dot{\cup }\cdots \dot{\cup }K_r$$[n]=K1??K2?????Kr be a partition of $$[n] = \{1,2,\ldots ,n\}$$[n]={1,2,?,n} and set $$\ell _i = |K_i|$$li=|Ki| for $$1\le i\le r$$1≤i≤r. Then the tuple $$P = \{K_1,K_2,\ldots ,K_r\}$$P={K1,K2,?,Kr} is an unordered partition of $$[n]$$[n] of shape $$[\ell _1,\ldots ,\ell _r]$$[l1,?,lr]. Let $${{\mathcal {P}}}$$P be the set of all partitions of $$[n]$$[n] of shape $$[\ell _1,\ldots ,\ell _r]$$[l1,?,lr]. Given a fixed shape $$[\ell _1,\ldots ,\ell _r]$$[l1,?,lr], we determine all subgroups $$G\le S_n$$G≤Sn that are transitive on $${{\mathcal {P}}}$$P in the following sense: Whenever $$P = \{K_1,\ldots ,K_r\}$$P={K1,?,Kr} and $$P' = \{K_1',\ldots ,K_r'\}$$P?={K1?,?,Kr?} are partitions of $$[n]$$[n] of shape $$[\ell _1,\ldots ,\ell _r]$$[l1,?,lr], there exists $$g\in G$$g?G such that $$g(P) = P'$$g(P)=P?, that is, $$\{g(K_1),\ldots ,g(K_r)\} = \{K_1',\ldots ,K_r'\}$${g(K1),?,g(Kr)}={K1?,?,Kr?}. Moreover, for an ordered shape, we determine all subgroups of $$S_n$$Sn that are transitive on the set of all ordered partitions of the given shape. That is, with $$P$$P and $$P'$$P? as above, $$g(K_i) = K_i'$$g(Ki)=Ki? for $$1\le i\le r$$1≤i≤r. As an application, we determine which Johnson graphs are Cayley graphs.

On the ensemble of optimal dominating and locating-dominating codes in a graph

Let G be a simple, undirected graph with vertex set V. For every v∈V, we denote by N(v) the set of neighbours of v, and let N[v]=N(v)∪{v}. A set C⊆V is said to be a dominating code in G if the sets N[v]∩C, v∈V, are all nonempty. A set C⊆V is said to be a locating-dominating code in G if the sets N[v]∩C, v∈V∖C, are all nonempty and distinct. The smallest size of a dominating (resp., locating-dominating) code in G is denoted by d(G) (resp., ℓ(G)).We study the ensemble of all the different optimal dominating (resp., locating-dominating) codes C, i.e., such that |C|=d(G) (resp., |C|=ℓ(G)) in a graph G, and strongly link this problem to that of induced subgraphs of Johnson graphs.

THE DISTINGUISHING NUMBERS OF MERGED JOHNSON GRAPHS

In present article, we determine the distinguishing number of the merged Johnson graphs which are generalization of both the Kneser graphs and of the Johnson graphs.

On Bounding the Bandwidth of Graphs with Symmetry

We derive a new lower bound for the bandwidth of a graph that is based on a new lower bound for the min-cut problem. Our new semidefinite programming relaxation of the min-cut problem is obtained by strengthening the known semidefinite programming relaxation for the quadratic assignment problem (or for the graph partition problem) by fixing two vertices in the graph; one on each side of the cut. Fixing results in several smaller subproblems that need to be solved to obtain the new bound. To efficiently solve these subproblems we exploit symmetry in the data; that is, both symmetry in the min-cut problem and symmetry in the graphs. To obtain upper bounds for the bandwidth of graphs with symmetry, we develop a heuristic approach based on the well-known reverse Cuthill–McKee algorithm, and that improves significantly its performance on the tested graphs. Our approaches result in the best known lower and upper bounds for the bandwidth of all graphs under consideration, i.e., Hamming graphs, 3-dimensional generalized Hamming graphs, Johnson graphs, and Kneser graphs, with up to 216 vertices.

On Sets with Few Intersection Numbers in Finite Projective and Affine Spaces

In this paper we study sets $X$ of points of both affine and projective spaces over the Galois field $\mathop{\rm{GF}}(q)$ such that every line of the geometry that is neither contained in $X$ nor disjoint from $X$ meets the set $X$ in a constant number of points and we determine all such sets. This study has its main motivation in connection with a recent study of neighbour transitive codes in Johnson graphs by Liebler and Praeger [Designs, Codes and Crypt., 2014]. We prove that, up to complements, in $\mathop{\rm{PG}}(n,q)$ such a set $X$ is either a subspace or $n=2,q$ is even and $X$ is a maximal arc of degree $m$. In $\mathop{\rm{AG}}(n,q)$ we show that $X$ is either the union of parallel hyperplanes or a cylinder with base a maximal arc of degree $m$ (or the complement of a maximal arc) or a cylinder with base the projection of a quadric. Finally we show that in the affine case there are examples (different from subspaces or their complements) in $\mathop{\rm{AG}}(n,4)$ and in $\mathop{\rm{AG}}(n,16)$ giving new neighbour transitive codes in Johnson graphs.

Binary Codes and Partial Permutation Decoding Sets from the Johnson Graphs

For $$k \ge 1$$k?1, $$n \ge 2k$$n?2k, the Johnson graph denoted by $$J(n,k),$$J(n,k), is the graph with vertex-set the set of all $$k$$k-subsets of $$\Omega = \{1, 2, \ldots , n\}$$Ω={1,2,?,n}, and any two vertices $$u$$u and $$v$$v are adjacent if and only if $$|u \cap v| = k-1$$|u?v|=k-1. In this paper the binary codes and their duals generated by an adjacency matrix of $$J(n,k)$$J(n,k) are described. The automorphism groups of the codes are determined, and by identifying suitable information sets, 3-PD-sets are determined for the code when $$k$$k is even.

On the number of matroids

We consider the problem of determining mn, the number of matroids on n elements. The best known lower bound on mn is due to Knuth (1974) who showed that log log mn is at least n − 3/2 log n−O(1). On the other hand, Piff (1973) showed that log log mn ≤ n − log n + log log n + O(1), and it has been conjectured since that the right answer is perhaps closer to Knuth's bound.We show that this is indeed the case, and prove an upper bound on log log mn that is within an additive 1 + o(1) term of Knuth's lower bound. Our proof is based on using some structural properties of non-bases in a matroid together with some properties of independent sets in the Johnson graph to give a compressed representation of matroids.

A problem of Shapozenko on Johnson graphs

The Johnson graph J(n,m) has the m-subsets of [n] as vertices and two subsets are adjacent in the graph if they share m−1 elements. Shapozenko asked about the isoperimetric function of Johnson graphs, that is, the cardinality of the smallest boundary of sets with k vertices in J(n,m) for each 1≤k≤(nm). We give an upper bound for the isoperimetric function of Johnson graphs. We show that, for given t and all sufficiently large n, the given bound is tight for k=(tm). We also show that the bound is tight for the small values of k≤m+1 and for all values of k when m=2.

Neighbour-transitive codes in Johnson graphs

The Johnson graph $$J(v,k)$$ has, as vertices, the $$k$$ -subsets of a $$v$$ -set $$\mathcal {V}$$ and as edges the pairs of $$k$$ -subsets with intersection of size $$k-1$$ . We introduce the notion of a neighbour-transitive code in $$J(v,k)$$ . This is a proper vertex subset $$\Gamma $$ such that the subgroup $$G$$ of graph automorphisms leaving $$\Gamma $$ invariant is transitive on both the set $$\Gamma $$ of ‘codewords’ and also the set of ‘neighbours’ of $$\Gamma $$ , which are the non-codewords joined by an edge to some codeword. We classify all examples where the group $$G$$ is a subgroup of the symmetric group $$\mathrm{Sym}\,(\mathcal {V})$$ and is intransitive or imprimitive on the underlying $$v$$ -set $$\mathcal {V}$$ . In the remaining case where $$G\le \mathrm{Sym}\,(\mathcal {V})$$ and $$G$$ is primitive on $$\mathcal {V}$$ , we prove that, provided distinct codewords are at distance at least $$3$$ , then $$G$$ is $$2$$ -transitive on $$\mathcal {V}$$ . We examine many of the infinite families of finite $$2$$ -transitive permutation groups and construct surprisingly rich families of examples of neighbour-transitive codes. A major unresolved case remains.

Deterministic random walks on finite graphs

ABSTRACTThe rotor‐router model, also known as the Propp machine, is a deterministic process analogous to a random walk on a graph. Instead of distributing tokens to randomly chosen neighbors, the Propp machine deterministically serves the neighbors in a fixed order by associating to each vertex a “rotor‐router” pointing to one of its neighbors. This paper investigates the discrepancy at a single vertex between the number of tokens in the rotor‐router model and the expected number of tokens in a random walk, for finite graphs in general. We show that the discrepancy is bounded by O (mn) at any time for any initial configuration if the corresponding random walk is lazy and reversible, where n and m denote the numbers of nodes and edges, respectively. For a lower bound, we show examples of graphs and initial configurations for which the discrepancy at a single vertex is Ω(m) at any time (> 0). For some special graphs, namely hypercube skeletons and Johnson graphs, we give a polylogarithmic upper bound, in terms of the number of nodes, for the discrepancy. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 46,739–761, 2015

On the fractional metric dimension of graphs

In Arumugam et al. (2013), Arumugam et al. studied the fractional metric dimension of the Cartesian product of two graphs, and proposed four open problems. In this paper, we determine the fractional metric dimension of vertex-transitive graphs, in particular, the fractional metric dimension of a vertex-transitive, distance-regular graph is expressed in terms of its intersection numbers. As an application, we calculate the fractional metric dimension of Hamming graphs and Johnson graphs, respectively. Moreover, we give an inequality for metric dimension and fractional metric dimension of an arbitrary graph, and determine all graphs for which the equality holds. Finally, we establish bounds on the fractional metric dimension of the Cartesian product of graphs. As a result, we completely solve the four open problems.

Embedding of eigenfunctions of the Johnson graph into eigenfunctions of the Hamming graph

Under study is the relationship between the eigenfunctions of the Johnson and Hamming graphs. An eigenfunction of a graph is an eigenvector of its adjacency matrix with some eigenvalue; moreover, an eigenfunction can be identically zero. We find a criterion for the embeddability of an eigenfunction of the Johnson graph J(n, w) with a given eigenvalue into a certain eigenfunction of the Hamming graph with a given eigenvalue.

Coloring hypercomplete and hyperpath graphs

Given a graph G with an induced subgraph H and a family F of graphs, we introduce a (hyper)graph HH(G;F)=(VH, EH), the hyper-H (hyper)graph of G with respect to F, whose vertices are induced copies of H in G, and \{H1,H2,\ldots,Hr\} \in EH if and only if the induced subgraph of G by the set \cupi=1r Hi is isomorphic to a graph F in the family F, and the integer r is the least integer for F with this property. When H is a k-complete or a k-path of G, we abbreviate HKk(G;F) and HPk(G;F) to Hk(G;F) and HPk(G;F), respectively. Our motivation to introduce this new (hyper)graph operator on graphs comes from the fact that the graph Hk(Kn;\{K2k\}) is isomorphic to the ordinary Kneser graph K(n;k) whenever 2k \leq n. As a generalization of the Lovasz--Kneser theorem, we prove that c(Hk(G;\{K2k\}))=c(G)-2k+2 for any graph G with w(G)=c(G) and any integer k\leq \lfloor w(G)/2\rfloor. We determine the clique and fractional chromatic numbers of Hk(G;\{K2k\}), and we consider the generalized Johnson graphs Hr(H;\{Kr+1\}) and show that c(Hr(H;\{Kr+1\}))\leq c(H) for any graph H and any integer r< w(H). By way of application, we construct examples of graphs such that the gap between their chromatic and fractional chromatic numbers is arbitrarily large. We further analyze the chromatic number of hyperpath (hyper)graphs HPk(G;Pm), and we provide upper bounds when m=k+1 and m=2k in terms of the k-distance chromatic number of the source graph.

Characterization of isometric embeddings of Grassmann graphs

Abstract Let V be an n-dimensional left vector space over a division ring R. We write Gk(V ) for the Grassmannian formed by k-dimensional subspaces of V and denote by Γk(V ) the associated Grassmann graph. Let also V′ be an n0-dimensional left vector space over a division ring R0. Isometric embeddings of Γk(V ) in Γk′(V′) are classified in [13]. A classification of J(n; k)-subsets in Gk′(V′), i.e. the images of isometric embeddings of the Johnson graph J(n; k) in Γk′(V′), is presented in [12]. We characterize isometric embeddings of Γk(V ) in Γk′(V′) as mappings which transfer apartments of Gk(V) to J(n; k)-subsets of Gk′(V′). This is a generalization of the earlier result concerning apartment preserving mappings [11, Theorem 3.10].