In this lecture we study the explicit constructions of expander graphs. Although we can construct expanders probabilistically, this does not suffice for many applications. • One applications of expander graphs is for reducing the randomness complexity of algorithms (cmp. Lecture 6), thus constructing the graph itself randomly does not serve this purpose. • Sometimes we may even need expanders o...

‎let $r$ be a commutative ring with non-zero identity. ‎we describe all $c_3$‎- ‎and $c_4$-free intersection graph of non-trivial ideals of $r$ as well as $c_n$-free intersection graph when $r$ is a reduced ring. ‎also, ‎we shall describe all complete, ‎regular and $n$-claw-free intersection graphs. ‎finally, ‎we shall prove that almost all artin rings $r$ have hamiltonian intersection graphs. ...

We study systematically the Lax description of the KdV hierarchy in terms of an operator which is the geometrical recursion operator. We formulate the Lax equation for the n-th flow, construct the Hamiltonians which lead to commuting flows. In this formulation, the recursion relation between the conserved quantities follows naturally. We give a simple and compact definition of all the Hamiltoni...

In this paper, we define the non-centralizer graph associated to a finite group G, as the graph whose vertices are the elements of G, and whose edges are obtained by joining two distinct vertices if their centralizers are not equal. We denote this graph by ΥG. The non-centralizer graph is used to study the properties of the non-commuting graph of an AC-group. We prove that the non-centralizer g...

The commuting graph of a group G, denoted by Γ(G), is a simple graph whose vertices are all non-central elements of G and two distinct vertices x, y are adjacent if xy = yx. In [1] it is conjectured that if M is a simple group and G is a group satisfying Γ(G) ∼= Γ(M), then G ∼= M . In this paper we prove this conjecture for many simple groups.

let r be a fnite commutative ring and n(r) be the set of non unit elements of r. the non unit graph of r, denoted by gamma(r), is the graph obtained by setting all the elements of n(r) to be the vertices and defning distinct vertices x and y to be adjacent if and only if x - yin n(r). in this paper, the basic properties of gamma(r) are investigated and some characterization results regarding co...

a graph $g$ is called a fractional‎ ‎$(k,n',m)$-critical deleted graph if any $n'$ vertices are removed‎ ‎from $g$ the resulting graph is a fractional $(k,m)$-deleted‎ ‎graph‎. ‎in this paper‎, ‎we prove that for integers $kge 2$‎, ‎$n',mge0$‎, ‎$nge8k+n'+4m-7$‎, ‎and $delta(g)ge k+n'+m$‎, ‎if‎ ‎$$|n_{g}(x)cup n_{g}(y)|gefrac{n+n'}{2}$$‎ ‎for each pair of non-adjac...

The Gruenberg-Kegel graph $\Gamma(G)$ associated with a finite group $G$ has as vertices the prime divisors of $|G|$, an edge from $p$ to $q$ if and only contains element order $pq$. This been subject much recent interest; one our goals here is give survey some this material, relating groups same graph. However, main aim prove several new results. Among them are following. - There infinitely ma...

In contrast to the determinant, no algorithm is known for exact determination of permanent a square matrix that runs in time polynomial its dimension. Consequently, non interacting fermions are classically efficiently simulatable while non-interacting bosons not, underpinning quantum supremacy arguments sampling output distribution photon interferometer arrays. This work introduces graph-theore...