We study geometric presentations of braid groups for particles that are constrained to move on a graph, i.e. network consisting nodes and edges. Our proposed set generators consists exchanges pairs junctions the graph certain circular moves where one particle travels around simple cycle graph. point out so defined often do not satisfy braiding relation known from 2D physics. accomplish full des...

We study the relationship between hyperfiniteness and problems in Borel graph combinatorics by adapting game-theoretic techniques introduced by Marks to the hyperfinite setting. We compute the possible Borel chromatic numbers and edge chromatic numbers of bounded degree acyclic hyperfinite Borel graphs and use this to answer a question of Kechris and Marks about the relationship between Borel c...

Let G be a rank 2 complete affine or hyperbolic Kac–Moody group over a finite field k. Then G is locally compact and totally disconnected. Let B− = HU− be the negative minimal parabolic subgroup of G, where H is the analog of a diagonal subgroup and U− is generated by all negative real root groups. Let w1 and w2 be the generators of the Weyl group. Let P − 1 = B − t Bw1B be the negative standar...

Thin Lie algebras are graded Lie algebras L = L ∞ i=1 Li with dimLi ≤ 2 for all i, and satisfying a more stringent but natural narrowness condition modeled on an analogous condition for pro-p groups. The two-dimensional homogeneous components of L, which include L1, are named diamonds. Infinite-dimensional thin Lie algebras with various diamond patterns have been produced, over fields of positi...

We introduce the active partition of ground set an oriented matroid perspective (or morphism, or quotient, strong map) on a linearly ordered set. The reorientations obtained by arbitrarily reorienting parts share same partition. This yields equivalence relation for perspective, whose classes are enumerated coefficients Tutte polynomial, and remarkable into Boolean lattices, from which we get sh...

Structural codes vis-a-vis structural counts, like polynomials of a molecular graph, are important in computing graph-theoretical descriptors which are commonly known as topological indices. These indices are most important for characterizing carbon nanotubes (CNTs). In this paper we have computed Sadhana index (Sd) for phenylenes and their hexagonal squeezes using structural codes (counts). Sa...

Cartesian products of complete graphs are known as Hamming graphs. Using embeddings into Cartesian products of quotient graphs we characterize subgraphs, induced subgraphs, and isometric subgraphs of Hamming graphs. For instance, a graph G is an induced subgraph of a Hamming graph if and only if there exists a labeling of E(G) fulfilling the following two conditions: (i) incident edges receive ...

The Baumslag-Solitar group $BS(2,3)$, is a so-called non-Hopfian group, meaning that it has an epimorphism $\phi$ onto itself, not injective. In particular this equivalent to saying $BS(2,3)$ non-trivial quotient isomorphic itself. As consequence the Cayley graph of itself up change generators. We describe on graph-level and take closer look at most common $\phi$. show its kernel free infinite ...

Let G be a finite simple graph on the vertex set $$V(G) = \{x_{1}, \ldots , x_{n}\}$$ and match(G), min-match(G) ind-match(G) matching number, minimum number induced of G, respectively. $$K[V(G)] K[x_{1}, x_{n}]$$ denote polynomial ring over field K $$I(G) \subset K[V(G)]$$ edge ideal G. The relationship between these graph-theoretic invariants ring-theoretic quotient K[V(G)]/I(G) has been stud...

In [M. Klin, A. Munemusa, M. Muzychuk, P.-H. Zieschang Directed strongly regular graphs obtained from coherent algebras. Linear Algebra and its Applications 337, (2004) 83-109] the flag algebra of a given balanced incomplete block design with parameters (ν, b, r, k, λ) where λ = 1, has been constructed. In this paper, we consider the association scheme which is related to this flag algebra. By ...