Error Graphs and the Reconstruction of Elements in Groups

Packing and covering problems for metric spaces, and graphs in particular, are of essential interest in combinatorics and coding theory. They are formulated in terms of metric balls of vertices. We consider a new problem in graph theory which is also based on the consideration of metric balls of vertices, but which is distinct from the traditional packing and covering problems. This problem is motivated by applications in information transmission when redundancy of messages is not sufficient for their exact reconstruction, and applications in computational biology when one wishes to restore an evolutionary process. It can be defined as the reconstruction, or identification, of an unknown vertex in a given graph from a minimal number of vertices (erroneous or distorted patterns) in a metric ball of a given radius r around the unknown vertex. For this problem it is required to find minimum restrictions for such a reconstruction to be possible and also to find efficient reconstruction algorithms under such minimal restrictions. In this paper we define error graphs and investigate their basic properties. A particular class of error graphs occurs when the vertices of the graph are the elements of a group, and when the path metric is determined by a suitable set of group elements. These are the undirected Cayley graphs. Of particular interest is the transposition Cayley graph on the symmetric group which occurs in connection with the analysis of transpositional mutations in molecular biology. We obtain a complete solution of the above problems for the transposition Cayley graph on the symmetric group.Comment: Journal of Combinatorial Theory A 200

Reconstruction of permutations distorted by single transposition errors

The reconstruction problem for permutations on $n$ elements from their erroneous patterns which are distorted by transpositions is presented in this paper. It is shown that for any $n \geq 3$ an unknown permutation is uniquely reconstructible from 4 distinct permutations at transposition distance at most one from the unknown permutation. The {\it transposition distance} between two permutations is defined as the least number of transpositions needed to transform one into the other. The proposed approach is based on the investigation of structural properties of a corresponding Cayley graph. In the case of at most two transposition errors it is shown that $\frac32(n-2)(n+1)$ erroneous patterns are required in order to reconstruct an unknown permutation. Similar results are obtained for two particular cases when permutations are distorted by given transpositions. These results confirm some bounds for regular graphs which are also presented in this paper.Comment: 5 pages, Report of paper presented at ISIT-200

Metric intersection problems in Cayley graphs and the Stirling recursion

In the symmetric group Sym(n) with n at least 5 let H be a conjugacy class of elements of order 2 and let \Gamma be the Cayley graph whose vertex set is the group G generated by H (so G is Sym(n) or Alt(n)) and whose edge set is determined by H. We are interested in the metric structure of this graph. In particular, for g\in G let B_{r}(g) be the metric ball in \Gamma of radius r and centre g. We show that the intersection numbers \Phi(\Gamma; r, g):=|\,B_{r}(e)\,\cap\,B_{r}(g)\,| are generalized Stirling functions in n and r. The results are motivated by the study of error graphs and related reconstruction problems.Comment: 18 page

Fuzzy transformations and extremality of Gibbs measures for the Potts model on a Cayley tree

We continue our study of the full set of translation-invariant splitting Gibbs measures (TISGMs, translation-invariant tree-indexed Markov chains) for the $q$-state Potts model on a Cayley tree. In our previous work \cite{KRK} we gave a full description of the TISGMs, and showed in particular that at sufficiently low temperatures their number is $2^{q}-1$. In this paper we find some regions for the temperature parameter ensuring that a given TISGM is (non-)extreme in the set of all Gibbs measures. In particular we show the existence of a temperature interval for which there are at least $2^{q-1} + q$ extremal TISGMs. For the Cayley tree of order two we give explicit formulae and some numerical values.Comment: 44 pages. To appear in Random Structures and Algorithm

Algebraic matroids with graph symmetry

This paper studies the properties of two kinds of matroids: (a) algebraic matroids and (b) finite and infinite matroids whose ground set have some canonical symmetry, for example row and column symmetry and transposition symmetry. For (a) algebraic matroids, we expose cryptomorphisms making them accessible to techniques from commutative algebra. This allows us to introduce for each circuit in an algebraic matroid an invariant called circuit polynomial, generalizing the minimal poly- nomial in classical Galois theory, and studying the matroid structure with multivariate methods. For (b) matroids with symmetries we introduce combinatorial invariants capturing structural properties of the rank function and its limit behavior, and obtain proofs which are purely combinatorial and do not assume algebraicity of the matroid; these imply and generalize known results in some specific cases where the matroid is also algebraic. These results are motivated by, and readily applicable to framework rigidity, low-rank matrix completion and determinantal varieties, which lie in the intersection of (a) and (b) where additional results can be derived. We study the corresponding matroids and their associated invariants, and for selected cases, we characterize the matroidal structure and the circuit polynomials completely

Bipartite entangled stabilizer mutually unbiased bases as maximum cliques of Cayley graphs

We examine the existence and structure of particular sets of mutually unbiased bases (MUBs) in bipartite qudit systems. In contrast to well-known power-of-prime MUB constructions, we restrict ourselves to using maximally entangled stabilizer states as MUB vectors. Consequently, these bipartite entangled stabilizer MUBs (BES MUBs) provide no local information, but are sufficient and minimal for decomposing a wide variety of interesting operators including (mixtures of) Jamiolkowski states, entanglement witnesses and more. The problem of finding such BES MUBs can be mapped, in a natural way, to that of finding maximum cliques in a family of Cayley graphs. Some relationships with known power-of-prime MUB constructions are discussed, and observables for BES MUBs are given explicitly in terms of Pauli operators.Comment: 8 pages, 1 figur