Algebra and Discrete Mathematics. - № 2 (26). - 2018http://hdl.handle.net/123456789/43842024-08-06T04:23:27Z2024-08-06T04:23:27ZOn unicyclic graphs of metric dimension 2 with vertices of degree 4Dudenko, M.liynyk, B.O.http://hdl.handle.net/123456789/45552020-01-08T15:07:04Z2018-01-01T00:00:00ZOn unicyclic graphs of metric dimension 2 with vertices of degree 4
Dudenko, M.; liynyk, B.O.
We show that if G is a unicyclic graph with metric dimension 2 and {a, b} is a metric basis of G then the degree of any vertex v of G is at most 4 and degrees of both a and b are at most 2. The constructions of unispider and semiunispider graphs and their knittings are introduced. Using these constructions all unicyclic graphs of metric dimension 2 with vertices of degree 4 are characterized.
Dudenko M. On unicyclic graphs of metric dimension 2 with vertices of degree 4 / M. Dudenko, B.O liynyk // Algebra and Discrete Mathematics. - 2018. - Vol. 26. - Number 2. - Рp.256-269
2018-01-01T00:00:00ZAbelian doppelsemigroupsZhuchok, A.V.Knauer, K.http://hdl.handle.net/123456789/45542020-01-08T15:07:07Z2018-01-01T00:00:00ZAbelian doppelsemigroups
Zhuchok, A.V.; Knauer, K.
A doppelsemigroup is an algebraic system consisting of a set with two binary associative operations satisfying certain identities. Doppelsemigroups are a generalization of semi-groups and they have relationships with such algebraic structures as doppelalgebras, duplexes, interassociative semigroups, restrictive bisemigroups, dimonoids and trioids. This paper is devoted to the study of abelian doppelsemigroups. We show that every abelian doppelsemigroup can be constructed from a left and right commutative semigroup and describe the free abelian doppelsemigroup. We also characterize the least abelian congruence on the free doppel semigroup, give examples of abelian doppelsemigroups and find conditions under which the operations of an abelian doppelsemi group coincide.
Zhuchok A.V. Abelian doppelsemigroups / A.V.Zhuchok, K.Knauer // Algebra and Discrete Mathematics. - 2018. - Vol. 26. - Number 2. - Рp.290-304
2018-01-01T00:00:00ZSpectral properties of partial automorphisms of a binary rooted treeKochubinska, E.http://hdl.handle.net/123456789/45532020-01-08T15:07:04Z2018-01-01T00:00:00ZSpectral properties of partial automorphisms of a binary rooted tree
Kochubinska, E.
We study asymptotics of the spectral measure of a randomly chosen partial automorphism of a rooted tree. To every partial automorphism x we assign its action matrix Ax. It is shown that the uniform distribution on eigenvalues of Ax converges weakly in probability to δ0 as n → ∞, where δ0 is the delta measure concentrated at 0.
Kochubinska E. Spectral properties of partial automorphisms of a binary rooted tree / E.Kochubinska // Algebra and Discrete Mathematics. - 2018. - Vol. 26. - Number 2. - Рp. 280-289
2018-01-01T00:00:00ZConnectedness of spheres in Cayley graphsBrieussel, J.Gournay, A.http://hdl.handle.net/123456789/45522020-01-08T15:07:18Z2018-01-01T00:00:00ZConnectedness of spheres in Cayley graphs
Brieussel, J.; Gournay, A.
We introduce the notion of connection thickness of spheres in a Cayley graph, related to dead-ends and their retreat depth. It was well-known that connection thickness is bounded for finitely presented one-ended groups. We compute that for natural generating sets of lamplighter groups on a line or on a tree, connection thickness is linear or logarithmic respectively. We show that it depends strongly on the generating set. We give an example where the metric induced at the (finite) thickness of connection gives diameter of order n2 to the sphere of radius n. We also discuss the rarity of dead-ends and the relationships of connection thickness with cut sets in percolation theory and with almost-convexity. Finally, we present a list of open questions about spheres in Cayley graphs.
Brieussel J. Connectedness of spheres in Cayley graphs / J.Brieussel , A.Gournay // Algebra and Discrete Mathematics. - 2018. - Vol. 26. - Number 2. - Рp. 190-246
2018-01-01T00:00:00Z