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Some results on the co-intersection graph of submodules of a module

Lotf Ali Mahdavi, Yahya Talebi (2018)

Commentationes Mathematicae Universitatis Carolinae

Let R be a ring with identity and M be a unitary left R -module. The co-intersection graph of proper submodules of M , denoted by Ω ( M ) , is an undirected simple graph whose vertex set V ( Ω ) is a set of all nontrivial submodules of M and two distinct vertices N and K are adjacent if and only if N + K ≠ M . We study the connectivity, the core and the clique number of Ω ( M ) . Also, we provide some conditions on the module M , under which the clique number of Ω ( M ) is infinite and Ω ( M ) is a planar graph. Moreover, we give several...

Some results on the recognizability of the linear groups over the binary field

Mohammad Reza Darafsheh, Yaghoub Farjami, M. Khademi, Ali Reza Moghaddamfar (2005)

Commentationes Mathematicae Universitatis Carolinae

In this paper, we first find the set of orders of all elements in some special linear groups over the binary field. Then, we will prove the characterizability of the special linear group PSL ( 13 , 2 ) using only the set of its element orders.

Special m-hyperidentities in biregular leftmost graph varieties of type (2,0)

Apinant Anantpinitwatna, Tiang Poomsa-ard (2009)

Discussiones Mathematicae - General Algebra and Applications

Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type (2,0). We say that a graph G satisfies a term equation s ≈ t if the corresponding graph algebra A ( G ) ̲ satisfies s ≈ t. A class of graph algebras V is called a graph variety if V = M o d g Σ where Σ is a subset of T(X) × T(X). A graph variety V ' = M o d g Σ ' is called a biregular leftmost graph variety if Σ’ is a set of biregular leftmost term equations. A term equation s ≈ t is called an identity in a variety...

Stable rational cohomology of automorphism groups of free groups and the integral cohomology of moduli spaces of graphs.

Craig A. Jensen (2002)

Publicacions Matemàtiques

It is not known whether or not the stable rational cohomology groups H*(Aut(F∞);Q) always vanish (see Hatcher in [5] and Hatcher and Vogtmann in [7] where they pose the question and show that it does vanish in the first 6 dimensions). We show that either the rational cohomology does not vanish in certain dimensions, or the integral cohomology of a moduli space of pointed graphs does not stabilize in certain other dimensions. Similar results are stated for groups of outer automorphisms. This yields...

Structure of geodesics in the Cayley graph of infinite Coxeter groups

Ryszard Szwarc (2003)

Colloquium Mathematicae

Let (W,S) be a Coxeter system such that no two generators in S commute. Assume that the Cayley graph of (W,S) does not contain adjacent hexagons. Then for any two vertices x and y in the Cayley graph of W and any number k ≤ d = dist(x,y) there are at most two vertices z such that dist(x,z) = k and dist(z,y) = d - k. Allowing adjacent hexagons, but assuming that no three hexagons can be adjacent to each other, we show that the number of such intermediate vertices at a given distance from x and y...

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