Journal of Algebraic Structures and Their Applications
Volume:1 Issue: 1, Winter - Spring 2014

Let G be a group. The order graph of G is the (undirected) graph Γ(G), those whose vertices are non-trivial subgroups of G and two distinct vertices H and K are adjacent if and only if either o(H)|o(K) or o(K)|o(H). In this paper, we investigate the interplay between the group-theoretic properties of G and the graph-theoretic properties of Γ(G). For a finite group G, we show that Γ(G) is a connected graph with diameter at mosttwo, and Γ(G) is a complete graph if and only if G is a p-group for some prime number p. Furthermore, it is shown that Γ(G)=K5 if and only if either G≅Cp5,C3×C3, C2×C4 or G≅Q8.

We study a new class of Hv -structures called Fundamentally Very Thin. This is an extension of the well known class of the Very Thin hyperstructures. We present applications of these hyperstructures.

Let K be a field of characteristic p>0, K[[x]], the ring of formal power series over K, K((x)), the quotient field of K[[x]], and K(x) the field of rational functions over K. We shall give some characterizations of an algebraic function f∈K((x)) over K. Let L be a field of characteristic zero. The power series f∈L[[x]] is called differentially algebraic, if it satisfies a differential equation of the form P(x,y,y′,...)=0, where P is a non-trivial polynomial. This notion is defined over fields of characteristic zero and is not so significant over fields of characteristic p>0, since f(p)=0. We shall define an analogue of the concept of a differentially algebraic power series over K and we shall find some more related results.

In this paper, we introduce the concepts of right, left and product stabilizers on hoops and study some properties and the relation between them. And we try to find that how they can be equal and investigate that under what condition they can be filter, implicative filter, fantastic and positive implicative filter. Also, we prove that right and product stabilizers are filters and if they are proper, then they are prime filters. Then by using the right stabilizers produce a basis for a topology on hoops. We show that the generated topology by this basis is Baire, connected, locally connected and separable and we investigate the other properties of this topology. Also, by the similar way, we introduce the right, left and product stabilizers on quotient hoops and introduce the quotient topology that is generated by them and investigate that under what condition this topology is Hausdorff space, T0 or T1 spaces.

H\"{o}lder in 1893 characterized all groups of order pqr where p>q>r are prime numbers. In this paper, by using new presentations of these groups, we compute their full automorphism group.

Cospectrality of two graphs measures the differences between the ordered spectrum of these graphs in various ways. Actually,
the origin of this concept came back to Richard Brualdi's problems that are proposed in cite{braldi}:Let Gn and G′n be two nonisomorphic simple graphs on n vertices with spectra
lambda1geqlambda2geqcdotsgeqlambdan;;;textand;;;lambda′1geqlambda′2geqcdotsgeqlambda′n,
respectively. Define the distance between the spectra of Gn and G′n as
lambda(Gn,G′n)=sumni=1(lambdai−lambda′i)2;;;big(textoruse;sumni=1|lambdai−lambda′i|big).
Define the cospectrality of Gn by
textcs(Gn)=minlambda(Gn,G′n);:;G′n;;textnotisomorphicto;Gn.
Let textcsn=maxtextcs(Gn);:;Gn;;textagraphon;n;textvertices.
Investigation of textcs(Gn) for special classes of graphs and finding a good upper bound on textcsn are two main questions in this
subject.
In this paper, we briefly give some important results in this direction and then we collect all cospectrality measures of graphs with at most six vertices with respect to three norms. Also, we give the shape of all graphs that are closest (with respect to cospectrality measure) to a given graph G.