Related papers: Bipartite $q$-Kneser graphs and two-generated irre…

Let $f(x)\in \mathbb{F}_q[x]$ be an irreducible polynomial of degree $m$ and exponent $e$, and $n$ be a positive integer such that $\nu_p(q-1)\ge \nu_{p}(e)+\nu_p(n)$ for all $p$ prime divisor of $n$. We show a fast algorithm to determine…

In this paper we study prime graphs of finite groups. The prime graph of a finite group $G$, also known as the Gruenberg-Kegel graph, is the graph with vertex set {primes dividing $|G|$} and an edge $p$-$q$ if and only if there exists an…

We consider a 2-homogeneous bipartite distance-regular graph $\Gamma$ with diameter $D \geq 3$. We assume that $\Gamma$ is not a hypercube nor a cycle. We fix a $Q$-polynomial ordering of the primitive idempotents of $\Gamma$. This…

Our goal is to investigate a close relative of the independent transversal problem in the class of infinite $K_n$-free graphs: we show that for any infinite $K_n$-free graph $G=(V,E)$ and $m\in \mathbb N$ there is a minimal $r=r(G,m)$ such…

We use a tensor unfolding technique to prove a new identifiability result for discrete bipartite graphical models, which have a bipartite graph between an observed and a latent layer. This model family includes popular models such as…

Let $G$ be a finite group, and write ${\rm cd}(G)$ for the degree set of the complex irreducible characters of $G$. The group $G$ is said to satisfy the {\it two-prime hypothesis} if, for any distinct degrees $a, b \in {\rm cd}(G)$, the…

We introduce a quadratic form $Q$ on the space of functions on the gap poset $G$ of the numerical semigroup $\langle a,b\rangle$. We prove combinatorially that when evaluated on the indicator function of an upward closed subset $D$, this…

We define totally-isotropic polynomials of alternating matrix spaces over finite fields, by analogy with independence polynomials of graphs. Our main result shows that totally-isotropic polynomials of graphical alternating matrix spaces…

Let $q$ be a power of a prime $p$, $G$ be a finite abelian group, where $p$ does not divide $|G|$,and let $n$ be a positive integer. In this paper we find a formula for the number of irreducible representations of $G$ of a given dimension…

Counting dominating sets in a graph $G$ is closely related to the neighborhood complex of $G$. We exploit this relation to prove that the number of dominating sets $d(G)$ of a graph is determined by the number of complete bipartite…

The commuting graph of a group $G$ is the graph whose vertices are the elements of $G$, two distinct vertices joined if they commute. Our purpose in this paper is twofold: we discuss the computational problem of deciding whether a given…

In this paper we study the (2,k)-generation of the finite classical groups SL(4,q), Sp(4,q), SU(4,q^2) and their projective images. Here k is the order of an arbitrary element of SL(2,q), subject to the necessary condition k>= 3. When q is…

Using the Fronsdal-Galindo formula for the exponential mapping from the quantum algebra $U_{p,q}(gl(2))$ to the quantum group $GL_{p,q}(2)$, we show how the $(2j+1)$-dimensional representations of $GL_{p,q}(2)$ can be obtained by…

The Gruenberg--Kegel graph (or the prime graph) $\Gamma(G)$ of a finite group $G$ is defined as follows. The vertex set of $\Gamma(G)$ is the set of all prime divisors of the order of $G$. Two distinct primes $r$ and $s$ regarded as…

We show that with high probability the random graph $G_{n, 1/2}$ has an induced subgraph of linear size, all of whose degrees are congruent to $r\pmod q$ for any fixed $r$ and $q\geq 2$. More generally, the same is true for any fixed…

It is well known that the proportion of pairs of elements of $\operatorname{SL}(n,q)$ which generate the group tends to $1$ as $q^n\to \infty$. This was proved by Kantor and Lubotzky using the classification of finite simple groups. We give…

We give the first analysis of the computational complexity of {\it coalition structure generation over graphs}. Given an undirected graph $G=(N,E)$ and a valuation function $v:2^N\rightarrow\RR$ over the subsets of nodes, the problem is to…

Consider the random process in which the edges of a graph $G$ are added one by one in a random order. A classical result states that if $G$ is the complete graph $K_{2n}$ or the complete bipartite graph $K_{n,n}$, then typically a perfect…

We prove: (1) The group of multipliers of similitudes of a 12-dimensional anisotropic quadratic form over a field K with trivial discriminant and split Clifford invariant is generated by norms from quadratic extensions E/K such that q_E is…

Let $n$ and $k$ be integers with $n>2k, k\geq1$. We denote by $H(n, k)$ the $bipartite\ Kneser\ graph$, that is, a graph with the family of $k$-subsets and ($n-k$)-subsets of $[n] = \{1, 2, ... , n\}$ as vertices, in which any two vertices…