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Zero-divisor graphs for semigroups of order 7

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Abstract

In this paper, we extend the results of DeMeyer and DeMeyer (J Algebra 283:190–198, 2005) that determine a set of sufficient conditions for a given graph to be the zero-divisor graph of a commutative semigroup, and use these to construct a larger set of graphs of this type. We then use these results to classify the connected graphs on six vertices, determining whether or not each is the zero-divisor graph of a commutative semigroup of order seven. To accomplish this, we give specific examples of graphs that can be easily classified using the extensions. For those graphs to which neither the original results nor the extensions apply, we provide a method by which we can classify them.

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Authors and Affiliations

  1. Mathematics Department Chair, William Mason High School, Mason, OH,  45040, USA

    Johnothon Sauer

  2. Department of Mathematics, The University of Cincinnati, Cincinnati, OH,  45221, USA

    Tara L. Smith

Authors
  1. Johnothon Sauer
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  2. Tara L. Smith
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Correspondence to Johnothon Sauer.

Additional information

Communicated by Jean-Eric Pin.

Appendix

Appendix

The following are the Cayley tables corresponding to the graphs listed in Table 1. Beginning with the vertex in the lower left corner of each graph and proceeding counter-clockwise, the vertices are labeled \(a-b-c-z-y-x\).

Table 1 24 Graphs of commutative semigroups
Full size table
$$\begin{aligned} \begin{array}{ccc} \begin{array}{l|llllll} (1,1)&{} a&{} b&{} c&{} x&{} y&{} z \\ \hline a&{} a&{} a&{} a&{} 0&{} 0&{} a \\ b&{} a&{} b&{} b&{} 0&{} 0&{} a \\ c&{} a&{} b&{} c&{} x&{} 0&{} a \\ x&{} 0&{} 0&{} x&{} x&{} 0&{} 0 \\ y&{} 0&{} 0&{} 0&{} 0&{} y&{} y \\ z&{} a&{} a&{} a&{} 0&{} y&{} z \end{array}&{}\quad \begin{array}{l|llllll} (1,2)&{} a&{} b&{} c&{} x&{} y&{} z \\ \hline a&{} 0&{} a&{} a&{} 0&{} 0&{} 0 \\ b&{} a&{} b&{} b&{} 0&{} 0&{} a \\ c&{} a&{} b&{} b&{} 0&{} x&{} a \\ x&{} 0&{} 0&{} 0&{} 0&{} x&{} 0 \\ y&{} 0&{} 0&{} x&{} x&{} y&{} x \\ z&{} 0&{} a&{} a&{} 0&{} x&{} 0 \\ \end{array} &{}\quad \begin{array}{l|llllll} (1,3)&{} a&{} b&{} c&{} x&{} y&{} z \\ \hline a&{} a&{} a&{} a&{} 0&{} 0&{} 0 \\ b&{} a&{} b&{} b&{} 0&{} z&{} z \\ c&{} a&{} b&{} c&{} 0&{} z&{} z \\ x&{} 0&{} 0&{} 0&{} x&{} x&{} 0 \\ y&{} 0&{} z&{} z&{} x&{} x&{} 0 \\ z&{} 0&{} z&{} z&{} 0&{} 0&{} 0 \\ \end{array}\\ \begin{array}{l|llllll} (1,4)&{} a&{} b&{} c&{} x&{} y&{} z \\ \hline a&{} 0&{} a&{} a&{} 0&{} 0&{} 0 \\ b&{} a&{} b&{} b&{} 0&{} 0&{} a \\ c&{} a&{} b&{} c&{} 0&{} 0&{} a \\ x&{} 0&{} 0&{} 0&{} 0&{} x&{} 0 \\ y&{} 0&{} 0&{} 0&{} x&{} y&{} x \\ z&{} 0&{} a&{} a&{} 0&{} x&{} 0 \end{array}&{}\quad \begin{array}{l|llllll} (1,5)&{} a&{} b&{} c&{} x&{} y&{} z \\ \hline a&{} a&{} a&{} a&{} 0&{} 0&{} 0 \\ b&{} a&{} a&{} a&{} 0&{} 0&{} x \\ c&{} a&{} a&{} a&{} 0&{} 0&{} x \\ x&{} 0&{} 0&{} 0&{} 0&{} 0&{} x \\ y&{} 0&{} 0&{} 0&{} 0&{} 0&{} x \\ z&{} 0&{} x&{} x&{} x&{} x&{} z \end{array}&{}\quad \begin{array}{l|llllll} (1,6)&{} a&{} b&{} c&{} x&{} y&{} z \\ \hline a&{} 0&{} a&{} a&{} 0&{} 0&{} 0 \\ b&{} a&{} b&{} b&{} 0&{} 0&{} a \\ c&{} a&{} b&{} c&{} x&{} 0&{} a \\ x&{} 0&{} 0&{} x&{} x&{} 0&{} 0 \\ y&{} 0&{} 0&{} 0&{} 0&{} y&{} y \\ z&{} 0&{} a&{} a&{} 0&{} y&{} y \end{array} \\ \begin{array}{l|llllll} (2,1)&{} a&{} b&{} c&{} x&{} y&{} z \\ \hline a&{} 0&{} z&{} z&{} 0&{} 0&{} 0 \\ b&{} z&{} b&{} b&{} 0&{} 0&{}z \\ c&{} z&{} b&{} b&{} 0&{} x&{} z \\ x&{} 0&{} 0&{} 0&{} 0&{} x&{} 0 \\ y&{} 0&{} 0&{} x&{} x&{} y&{}0 \\ z&{} 0&{} z&{} z&{} 0&{} 0&{} 0 \end{array}&{}\quad \begin{array}{l|llllll} (2,2)&{} a&{} b&{} c&{} x&{} y&{} z \\ \hline a&{} 0&{} a&{} a&{} 0&{} 0&{} 0 \\ b&{} a&{} b&{} b&{} 0&{} 0&{} z \\ c&{} a&{} b&{} b&{} 0&{} 0&{} z \\ x&{} 0&{} 0&{} 0&{} x&{} x&{} 0 \\ y&{} 0&{} 0&{} 0&{} x&{} x&{} 0 \\ z&{} 0&{} z&{} z&{} 0&{} 0&{} 0 \\ \end{array}&{}\quad \begin{array}{l|llllll} (2,3)&{} a&{} b&{} c&{} x&{} y&{} z \\ \hline a&{} a&{} 0&{} 0&{} 0&{} a&{} a \\ b&{} 0&{} b&{} b&{} 0&{} 0&{} 0 \\ c&{} 0&{} b&{} c&{} x&{} x&{} 0 \\ x&{} 0&{} 0&{} x&{} x&{} x&{} 0 \\ y&{} a&{} 0&{} x&{} x&{} y&{} a \\ z&{} a&{} 0&{} 0&{} 0&{} a&{} a \end{array}\\ \begin{array}{l|llllll} (2,4)&{} a&{} b&{} c&{} x&{} y&{} z \\ \hline a&{} a&{} a&{} a&{} 0&{} 0&{} 0 \\ b&{} a&{} a&{} a&{} 0&{} 0&{} 0 \\ c&{} a&{} a&{} a&{} 0&{} 0&{} x \\ x&{} 0&{} 0&{} 0&{} 0&{} 0&{} x \\ y&{} 0&{} 0&{} 0&{} 0&{} 0&{} x \\ z&{} 0&{} 0&{} x&{} x&{} x&{} z \end{array}&{}\quad \begin{array}{l|llllll} (2,5)&{} a&{} b&{} c&{} x&{} y&{} z \\ \hline a&{} a&{} a&{} a&{} 0&{} 0&{} 0 \\ b&{} a&{} b&{} a&{} 0&{} 0&{} 0 \\ c&{} a&{} a&{} c&{} 0&{} z&{} z \\ x&{} 0&{} 0&{} 0&{} x&{} x&{} 0 \\ y&{} 0&{} 0&{} z&{} x&{} x&{} 0 \\ z&{} 0&{} 0&{} z&{} 0&{} 0&{} 0 \\ \end{array}&{}\quad \begin{array}{l|llllll} (2,6)&{} a&{} b&{} c&{} x&{} y&{} z \\ \hline a&{} a&{} a&{} a&{} 0&{} 0&{} 0 \\ b&{} a&{} b&{} a&{} x&{} 0&{} 0 \\ c&{} a&{} a&{} c&{} 0&{} y&{} z \\ x&{} 0&{} x&{} 0&{} x&{} 0&{} 0 \\ y&{} 0&{} 0&{} y&{} 0&{} y&{} 0 \\ z&{} 0&{} 0&{} z&{} 0&{} 0&{} z \end{array} \end{array} \end{aligned}$$
$$\begin{aligned} \begin{array}{lll} \begin{array}{l|llllll} (3,1)&{} a&{} b&{} c&{} x&{} y&{} z \\ \hline a&{} a&{} a&{} a&{} 0&{} 0&{} 0 \\ b&{} a&{} a&{} a&{} 0&{} x&{} 0 \\ c&{} a&{} a&{} c&{} 0&{} 0&{} z \\ x&{} 0&{} 0&{} 0&{} 0&{} x&{} 0 \\ y&{} 0&{} x&{} 0&{} x&{} y&{} 0 \\ z&{} 0&{} 0&{} z&{} 0&{} 0&{} z \end{array}&{}\quad \begin{array}{l|llllll} (3,2)&{} a&{} b&{} c&{} x&{} y&{} z \\ \hline a&{} a&{} a&{} a&{} 0&{} 0&{} 0 \\ b&{} a&{} a&{} a&{} 0&{} 0&{} 0 \\ c&{} a&{} a&{} a&{} 0&{} 0&{} 0 \\ x&{} 0&{} 0&{} 0&{} 0&{} x&{} 0 \\ y&{} 0&{} 0&{} 0&{} x&{} y&{} z \\ z&{} 0&{} 0&{} 0&{} 0&{} z&{} z \\ \end{array}&{}\quad \begin{array}{l|llllll} (3,3)&{} a&{} b&{} c&{} x&{} y&{} z \\ \hline a&{} a&{} a&{} a&{} 0&{} 0&{} 0 \\ b&{} a&{} a&{} a&{} 0&{} 0&{} 0 \\ c&{} a&{} a&{} c&{} x&{} 0&{} 0 \\ x&{} 0&{} 0&{} x&{} x&{} 0&{} 0 \\ y&{} 0&{} 0&{} 0&{} 0&{} y&{} y \\ z&{} 0&{} 0&{} 0&{} 0&{} y&{} y \end{array}\\ \begin{array}{l|llllll} (3,4)&{} a&{} b&{} c&{} x&{} y&{} z \\ \hline a&{} 0&{} x&{} 0&{} 0&{} 0&{} y \\ b&{} x&{} b&{} x&{} x&{} 0&{} 0 \\ c&{} 0&{} x&{} 0&{} 0&{} 0&{} y \\ x&{} 0&{} x&{} 0&{} 0&{} 0&{} 0 \\ y&{} 0&{} 0&{} 0&{} 0&{} 0&{} y \\ z&{} y&{} 0&{} y&{} 0&{} y&{} z \end{array}&{}\quad \begin{array}{l|llllll} (3,5)&{} a&{} b&{} c&{} x&{} y&{} z \\ \hline a&{} a&{} a&{} 0&{} 0&{} 0&{} 0 \\ b&{} a&{} a&{} x&{} 0&{} 0&{} 0 \\ c&{} 0&{} x&{} c&{} x&{} 0&{} x \\ x&{} 0&{} 0&{} x&{} 0&{} 0&{} 0 \\ y&{} 0&{} 0&{} 0&{} 0&{} y&{} y \\ z&{} 0&{} 0&{} x&{} 0&{} y&{} y \end{array}&{}\quad \begin{array}{l|llllll} (3,6)&{} a&{} b&{} c&{} x&{} y&{} z \\ \hline a&{} 0&{} a&{} 0&{} 0&{} 0&{} 0 \\ b&{} a&{} b&{} a&{} 0&{} 0&{} a \\ c&{} 0&{} a&{} 0&{} 0&{} 0&{} 0 \\ x&{} 0&{} 0&{} 0&{} 0&{} x&{} 0 \\ y&{} 0&{} 0&{} 0&{} x&{} y&{} x \\ z&{} 0&{} a&{} 0&{} 0&{} x&{} 0 \end{array}\\ \begin{array}{l|llllll} (4,1)&{} a&{} b&{} c&{} x&{} y&{} z \\ \hline a&{} 0&{} a&{} 0&{} 0&{} 0&{} 0 \\ b&{} a&{} b&{} a&{} 0&{} a&{} a \\ c&{} 0&{} a&{} 0&{} 0&{} 0&{} 0 \\ x&{} 0&{} 0&{} 0&{} x&{} x&{} 0 \\ y&{} 0&{} a&{} 0&{} x&{} x&{} 0 \\ z&{} 0&{} a&{} 0&{} 0&{} 0&{} z \end{array}&{}\quad \begin{array}{l|llllll} (4,2)&{} a&{} b&{} c&{} x&{} y&{} z \\ \hline a&{} 0&{} a&{} 0&{} 0&{} 0&{}0 \\ b&{} a&{} b&{} 0&{} 0&{} 0&{} 0 \\ c&{} 0&{} 0&{} 0&{} 0&{} 0&{} c \\ x&{} 0&{} 0&{} 0&{} x&{} x&{} 0 \\ y&{} 0&{} 0&{} 0&{} x&{} x&{} c \\ z&{} 0&{} 0&{} c&{} 0&{} c&{} z \end{array}&{}\quad \begin{array}{l|llllll} (4,3)&{} a&{} b&{} c&{} x&{} y&{} z \\ \hline a&{} 0&{} a&{} 0&{} 0&{} 0&{}0 \\ b&{} a&{} b&{} 0&{} 0&{} 0&{} 0 \\ c&{} 0&{} 0&{} c&{} 0&{} 0&{} c \\ x&{} 0&{} 0&{} 0&{} x&{} 0&{} x \\ y&{} 0&{} 0&{} 0&{} 0&{} y&{} y \\ z&{} 0&{} 0&{} c&{} x&{} y&{} z \end{array}\\ \begin{array}{l|llllll} (4,4)&{} a&{} b&{} c&{} x&{} y&{} z \\ \hline a&{} a&{} a&{} 0&{} 0&{} 0&{} 0 \\ b&{} a&{} b&{} 0&{} x&{} 0&{} 0 \\ c&{} 0&{} 0&{} c&{} 0&{} 0&{} c \\ x&{} 0&{} x&{} 0&{} x&{} 0&{} 0 \\ y&{} 0&{} 0&{} 0&{} 0&{} y&{} y \\ z&{} 0&{} 0&{} c&{} 0&{} y&{} z \end{array}&{}\quad \begin{array}{l|llllll} (4,5)&{} a&{} b&{} c&{} x&{} y&{} z \\ \hline a&{} 0&{} a&{} 0&{} 0&{} 0&{} 0 \\ b&{} a&{} b&{} 0&{} 0&{} 0&{} 0 \\ c&{} 0&{} 0&{} 0&{} 0&{} 0&{} c \\ x&{} 0&{} 0&{} 0&{} 0&{} x&{} 0 \\ y&{} 0&{} 0&{} 0&{} x&{} y&{} 0 \\ z&{} 0&{} 0&{} c&{} 0&{} 0&{} z \end{array}&{}\quad \begin{array}{l|llllll} (4,6)&{} a&{} b&{} c&{} x&{} y&{} z \\ \hline a&{} a&{} a&{} a&{} 0&{} 0&{} 0 \\ b&{} a&{} a&{} a&{} 0&{} 0&{} x \\ c&{} a&{} a&{} b&{} 0&{} x&{} y \\ x&{} 0&{} 0&{} 0&{} 0&{} 0&{} x \\ y&{} 0&{} 0&{} x&{} 0&{} x&{} y \\ z&{} 0&{} x&{} y&{} x&{} y&{} z \\ \end{array} \end{array} \end{aligned}$$
Table 2 Two graphs that are not the graph of a commutative semigroup
Full size table

Finally, neither of the graphs in Table 2 is the graph of a commutative semigroup:

  • For graph N1, using the same labeling mentioned at the beginning of the appendix, \(b^{2}=b, z^{2}=z, ab\in \{a,~c\}\), and \(bc\in \{a,c\}\), but then bz does not have a value.

  • A complete treatment of graph N2 was given above.

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Sauer, J., Smith, T.L. Zero-divisor graphs for semigroups of order 7. Semigroup Forum 90, 155–173 (2015). https://doi.org/10.1007/s00233-014-9605-0

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