Related papers: On Matrix Product Factorization of Cayley graphs

Suppose G is a finite group, such that |G| = 27p, where p is prime. We show that if S is any generating set of G, then there is a hamiltonian cycle in the corresponding Cayley graph Cay(G;S).

Let $R$ be a commutative ring and let $U(R)$ be multiplicative group of unit elements of $R$. In 2012, Khashyarmanesh et al. defined generalized unit and unitary Cayley graph, $\Gamma(R, G, S)$, corresponding to a multiplicative subgroup…

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…

We present a systematic construction of F-theory compactifications with Abelian gauge symmetries in addition to a non-Abelian gauge group G. The formalism is generally applicable to models in global Tate form but we focus on the…

Let $G$ be a group and let $S$ be a generating set of $G$. In this article, we introduce a metric $d_C$ on $G$ with respect to $S$, called the cardinal metric. We then compare geometric structures of $(G, d_C)$ and $(G, d_W)$, where $d_W$…

Graph connectivity serves as a fundamental metric for evaluating the reliability and fault tolerance of interconnection networks. To more precisely characterize network robustness, the concept of cyclic connectivity has been introduced,…

We study F-theory compactifications with up to two Abelian gauge group factors that are based on elliptically fibered Calabi-Yau 4-folds describable as generic hypersurfaces. Special emphasis is put on elliptic fibrations based on generic…

These notes concern aspects of various graphs whose vertex set is a group $G$ and whose edges reflect group structure in some way (so that they are invariant under the action of the automorphism group of $G$). The graphs I will discuss are…

We characterize the equivalence and the weak equivalence of Cayley graphs for a finite group $\C{A}$. Using these characterizations, we find enumeration formulae of the equivalence classes and weak equivalence classes of Cayley graphs. As…

For a transitive infinite connected graph $G$, let $\mu(G)$ be its connective constant. Denote by $\mathbf{\cal G}$ the set of Cayley graphs for finitely generated infinite groups with an infinite-order generator which is independent of…

We introduce a notion of Ricci curvature for Cayley graphs that can be thought of as "medium-scale" because it is neither infinitesimal nor asymptotic, but based on a chosen finite radius parameter. We argue that it gives the foundation for…

A connected graph is called \emph{geodetic} if there is a unique shortest path between each pair of vertices. We introduce a systematic method for constructing new presentations of free products that give rise to previously unknown geodetic…

Let $S\subset \text{SL}_2(\mathbb Z)\times \text{SL}_2(\mathbb Z)$ or $\text{SL}_2(\mathbb Z)\ltimes \mathbb Z^2$ be finite symmetric and assume $S$ generates a group $G$ which is a Zariski-dense subgroup $\text{SL}_2(\mathbb Z)\times…

We show that a "mate'' $B$ of a set $A$ in a near-factorization $(A,B)$ of a finite group $G$ is unique. Further, we describe how to compute the mate $B$ very efficiently using an explicit formula for $B$. We use this approach to give an…

The theory of quantum symmetric pairs provides a universal K-matrix which is an analogue of the universal R-matrix for quantum groups. The main ingredient in the construction of the universal K-matrix is a quasi K-matrix which has so far…

Let T_1,..., T_d be homogeneous trees with degrees q_1+1,..., q_d+1>=3, respectively. For each tree, let h:T_j->Z be the Busemann function with respect to a fixed boundary point (end). Its level sets are the horocycles. The horocyclic…

A graph $\Gamma$ is said to be unstable if for the direct product $\Gamma \times K_2$, $Aut(\Gamma \times K_2)$ is not isomorphic to $Aut(\Gamma) \times \mathbb{Z}_2$. In this paper we show that a connected and non-bipartite Cayley graph…

We present some observations on a restricted variant of unitary Cayley graphs modulo n, and the implications for a decomposition of elements of symplectic operators over the integers modulo n. We define quadratic unitary Cayley graphs G_n,…

A Cayley graph of a group $H$ is a finite simple graph $\Gamma$ such that ${\rm Aut}(\Gamma)$ contains a subgroup isomorphic to $H$ acting regularly on $V(\Gamma)$, while a Haar graph of $H$ is a finite simple bipartite graph $\Sigma$ such…

In this paper, we investigate structural properties of the Cayley graph of a quandle and describe this graph for several important classes of quandles, including conjugation, Takasaki, dihedral, and Alexander quandles. In particular, we…