Related papers: Cubic Graphical Regular Representations of $\mathr…

Let $G_{\mathbb{Q}_p}$ be the absolute Galois group of $\mathbb{Q}_p$ and let $L$ be a finite extension of $\mathbb{Q}_p$. Moreover let $\bar\rho:G_{\mathbb{Q}_p}\rightarrow GL_n(k_L)$ be a continous representation of $G_{\mathbb{Q}_p}$,…

A graph is square-complementary (squco, for short) if its square and complement are isomorphic. We prove that there are no squco graphs with girth 6, that every bipartite graph is an induced subgraph of a squco bipartite graph, that the…

Let $p$ be an odd prime, and $D_{2p}=\langle a,b\mid a^p=b^2=1,bab=a^{-1}\rangle$ the dihedral group of order $2p$. In this paper, we completely classify the cubic Cayley graphs on $D_{2p}$ up to isomorphism by means of spectral method. By…

We produce a sequence of finite dimensional representations of the fundamental group $\pi_1(S)$ of a closed surface where all simple closed curves act with finite order, but where each non--simple closed curve eventually acts with infinite…

We describe a technique to determine the automorphism group of a geometrically represented graph, by understanding the structure of the induced action on all geometric representations. Using this, we characterize automorphism groups of…

Irreducible representations are the building blocks of general, semisimple Galois representations \rho, and cuspidal representations are the building blocks of automorphic forms \pi of the general linear group. It is expected that when an…

Let $G$ be a bridgeless cubic graph. Consider a list of $k$ 1-factors of $G$. Let $E_i$ be the set of edges contained in precisely $i$ members of the $k$ 1-factors. Let $\mu_k(G)$ be the smallest $|E_0|$ over all lists of $k$ 1-factors of…

An isomorphism between the group ring of a finite group and a ring of certain block diagonal matrices is established. The group ring $RG$ of a finite group $G$ is isomorphic to the set of {\em group ring matrices} over $R$. It is shown that…

A unit cube in $k$ dimensional space (or \emph{$k$-cube} in short) is defined as the Cartesian product $R_1\times R_2\times...\times R_k$ where $R_i$(for $1\leq i\leq k$) is a closed interval of the form $[a_i,a_i+1]$ on the real line. A…

In this paper, we examine the groups $G_2$ and $G_3$ associated to the $2 \times 2$ and $3 \times 3$ Rubik's cubes. We express $G_2$ and $G_3$ in terms of familiar groups and exhibit a split homomorphism $\psi: G_3 \longrightarrow G_2$ to…

A polygonal complex in euclidean 3-space is a discrete polyhedron-like structure with finite or infinite polygons as faces and finite graphs as vertex-figures, such that a fixed number r of faces surround each edge. It is said to be regular…

The reduced power graph $\mathcal{RP}(G)$ of a group $G$ is the graph with vertex set $G$ and two vertices $u$ and $v$ are adjacent if and only if $\left\langle v\right\rangle \subset \left\langle u \right\rangle $ or $\left\langle…

Let \pi(G) denote the set of prime divisors of the order of a finite group G. The prime graph of G is the graph with vertex set \pi(G) with edges {p,q} if and only if there exists an element of order pq in G. In this paper, we prove that a…

We prove that every 3-regular graph with no circuit of length less than six has a subgraph isomorphic to a subdivision of the Petersen graph.

Let $Q^+(2n+1,q)$ be a hyperbolic quadric of $\PG(2n+1,q)$. Fix a generator $\Pi$ of the quadric. Define $\cG_n$ as the graph with vertex set the points of $Q^+(2n+1,q)\setminus \Pi$ and two vertices adjacent if they either span a secant to…

We obtain an explicit formula for the quadratic embedding constant (QEC) of a strongly regular graph $\mathrm{srg}(n,k,\lambda,\mu)$ with $\mu\ge1$. By using QEC we give a necessary and sufficient condition for a strongly regular graph to…

A random algebraic graph is defined by a group $G$ with a uniform distribution over it and a connection $\sigma:G\longrightarrow[0,1]$ with expectation $p,$ satisfying $\sigma(g)=\sigma(g^{-1}).$ The random graph…

Given a finite group $G$ acting on a surface $S$, the centralizer of G in the mapping class group $\textrm{Mod}(S)$ has a natural representation given by its action on the homology $H_1(S; \mathbb{Q})$. We consider the question of whether…

Every Ree group $R(q)$, with $q\neq 3$ an odd power of 3, is the automorphism group of an abstract regular polytope, and any such polytope is necessarily a regular polyhedron (a map on a surface). However, an almost simple group $G$ with…

We give a proof, using so-called fusion rings and q-deformations of Brauer algebras that the representation ring of an orthogonal or symplectic group can be obtained as a quotient of a ring Gr(O(\infinity)). This is obtained here as a…