For the list object, introduced in Chapter 5, it was shown that each data element contains at most one predecessor element and one successor element. Therefore, for any given data element or node in the list structure, we can talk in terms of a next element and a previous element.

Graphs and Digraphs

This article gives a survey of all results on the power graphs of groups and semigroups obtained in the literature. Various conjectures due to other authors, questions and open problems are also included.

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Power graphs: A survey

Research on combinatorial properties of sequences in groups and semigroups originates from Bernhard Neumann's theorem answering a question of Paul Erd"{o}s. For results on related combinatorial properties of sequences in semigroups we refer to the book [3]. In 2000 the authors introduced a new combinatorial property and described all groups satisfying it. The present paper extends this result to all semigroups.

A combinatorial property and power graphs of semigroups

The structure and metric dimension of the power graph of a finite group

We describe semigroups satisfying a combinatorial property defined in terms of Cayley graphs.

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A combinatorial property and Cayley graphs of semigroups

Matrices provide essential tools in many branches of mathematics, and matrix semi-groups have applications in various areas. In this paper we give a complete descriptionof all infinite matrix semigroups satisfying a certain combinatorial property definedin terms of power graphs.Research on combinatorial properties of words in groups originates from the fol-lowing well-known theorem due to Bernhard Neumann [12], which was obtained as ananswer to a question of Paul Erdos: a group is centre-by-finite if and only if every infinitesequence contains a pair of elements that commute. Combinatorial properties of groupsand semigroups with all infinite subsets containing certain special elements have beenconsidered by Bell, Blyth, Curzio, de Luca, Gillam, Hall, Higgins, Justin, Longobardi,Maj, Okninski, Piochi, Pirillo, Restivo, Reutenauer, Rhemtulla, Robinson, Sapir, Shumy-atsky, Simon, Varricchio and other authors, and a survey of this direction was given bythe first author in [7] (see also [2, 3, 6, 11]).The following combinatorial property was introduced in [9] using power graphs. Thepower graph Pow(S) of a semigroup S has all elements of S as vertices, and it has edges(u, v) for all u,v € S such that u ^ v and v is a power of u. Let D be a directed graph.We say that an infinite semigroup 5 is power D-saturated if and only if, for every infinitesubset T of 5, the power graph of S has a subgraph isomorphic to D with all verticesin T. In this paper we describe all pairs (£>, S), where D is a directed graph and S is amatrix semigroup, such that S is power D-saturated.The reader is referred to [1, 4, 14] for standard graph, semigroup and group theoreticterminology, respectively. By the word 'graph' we mean a directed graph without loopsor multiple edges. A graph is said to be acyclic if it has no directed cycles.We refer to [10, 13] for preliminaries on fields and matrix semigroups, respectively.For a skew field K, the set of all n x n matrices with entries in K i

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/1FC3B1C1547328EA70C0ADC1F53975ED/S0004972700019390a.pdf/div-class-title-power-graphs-and-semigroups-of-matrices-div.pdf

R. Smolíková

Papers

Power graphs and semigroups of matrices

On fuzzy regular languages