Some properties on the lexicographic product of graphs obtained by monogenic semigroups
© Akgunes et al.; licensee Springer. 2013
Received: 6 February 2013
Accepted: 24 April 2013
Published: 13 May 2013
In (Das et al. in J. Inequal. Appl. 2013:44, 2013), a new graph on monogenic semigroups (with zero) having elements was recently defined. The vertices are the non-zero elements and, for , any two distinct vertices and are adjacent if in . As a continuing study, in an unpublished work, some well-known indices (first Zagreb index, second Zagreb index, Randić index, geometric-arithmetic index, atom-bond connectivity index, Wiener index, Harary index, first and second Zagreb eccentricity indices, eccentric connectivity index, the degree distance) over were investigated by the same authors of this paper.
In the light of the above references, our main aim in this paper is to extend these studies to the lexicographic product over . In detail, we investigate the diameter, radius, girth, maximum and minimum degree, chromatic number, clique number and domination number for the lexicographic product of any two (not necessarily different) graphs and .
MSC:05C10, 05C12, 06A07, 15A18, 15A36.
1 Introduction and preliminaries
The base of the graph is actually zero-divisor graphs (cf. ). In fact, the history of studying zero-divisor graphs began over commutative rings by the paper , and then it was followed over commutative and noncommutative rings by some of the joint papers [3–5]. After that the same terminology has been converted to commutative and noncommutative semigroups [6, 7].
Then, by following the definition given in , an undirected (zero-divisor) graph associated to was obtained as in the following. The vertices of the graph are labeled by the nonzero zero-divisors (in other words, all nonzero element) of , and any two distinct vertices and , where () are connected by an edge in the case with the rule if and only if . The fundamental spectral properties such as the diameter, girth, maximum and minimum degree, chromatic number, clique number, degree sequence, irregularity index and dominating number for this new graph are presented in . Furthermore, in an unpublished work, the same authors of this paper studied the first and second Zagreb indices, Randić index, geometric-arithmetic index and atom-bond connectivity index, Wiener index, Harary index, the first and second Zagreb eccentricity indices, eccentric connectivity index and the degree distance to indicate the importance of the graph .
In this paper, by considering , we present some certain results for the diameter, radius, girth, maximum and minimum degrees, and finally chromatic, clique and domination numbers.
2 Main results
It is known that the girth of a simple graph G is the length of the shortest cycle contained in that graph. However, if G does not contain any cycle, then the girth of it is assumed to be infinity. Thus the first theorem of this paper is the following.
as desired. □
The degree of a vertex v of G is the number of vertices adjacent to v. Among all degrees, the maximum (or the minimum ) degrees of G is the number of the largest (or the smallest) degree in G .
since the vertex is adjacent to all the other vertices.
On the other hand, let us take the vertex . Then, again by (3), the adjacency of with a vertex holds only if we have or and . That means the vertex is connected to and . Thus , as required. □
We then have the next result.
Hence the result. □
Hence, the eccentricity is equal to 1, which implies the required result. □
A subset D of the vertex set of a graph G is called a dominating set if every vertex is joined to at least one vertex of D by an edge. Additionally, the domination number is the number of vertices in the smallest dominating set for G. (We may refer to  for the fundamentals of a domination number.)
In our case, by (2), the dominating set is defined by since the vertices are adjacent to all the other vertices. Hence we obtain the next result.
Theorem 5 .
Basically, the coloring of G is to be an assignment of colors (elements of some set) to the vertices of G, one color to each vertex, so that adjacent vertices are assigned distinct colors. If n colors are used, then the coloring is referred to as an n-coloring. If there exists an n-coloring of G, then G is called n-colorable. The minimum number n for which G is n-colorable is called the chromatic number of G and is denoted by .
In addition, there exists another graph parameter, namely the clique of a graph G. In fact, depending on the vertices, each of the maximal complete subgraphs of G is called a clique. Moreover, the largest number of vertices in any clique of G is called the clique number and denoted by . In general, by , it is well known that for any graph G. For every induced subgraph H of G, if holds, then G is called a perfect graph .
By constructing the next result (see Theorem 6 below) for the chromatic number over the lexicographic product of the graphs and , we shall present a negative answer of a result given in  (see Remark 1).
We recall that for a real number r, the notation denotes the least integer ≥r. This fact will be needed for some of our results below.
The proof of the following lemma can be found in [, Theorem 6].
Lemma 1 ()
The next result is an extension of the above lemma to the lexicographic product.
In other words, .
Proof First step: The list of vertices that the vertex is adjacent to all the other vertices was given in (2). That means the color that was used for cannot be used for any other vertices. So, let us suppose that the color used for the vertex is labeled by . Secondly, if we consider the vertex , then it is easy to see that is adjacent to all the vertices except the vertex . Thus, the color for , say , can be also used only for . As a similar idea, the vertex is adjacent to all the vertices except the vertices and . Thus the color, say , for can be also used only for the vertex . (Notice that the color has been already used for in the previous step.) After that, following the same progress, we see that the total of different colors is needed to handle the coloring of all vertices in the set .
is labeled by .
Thus the color, say , for can be also used only for the vertices and . (Again, notice that the color has been already used for previously.)
as desired. □
Remark 1 It is clear that . This trivial upper bound is attained for any G and H with and . However, in Theorem 6, we obtained an equality between and . But it was shown in , that there is not any product ∗ of graphs for which the equality holds for all graphs G and H.
In [, Theorem 3.1], the authors proved that the clique number is preserved under the lexicographic product for any graphs G and H. In the following, we deal with this result by considering our special graphs. Before that, we need to present the following lemma, the truthfulness of which is quite clear.
i.e., for all i, j, a, b.
Hence we obtain , as required. □
which implies that the lexicographic product preserves the perfectness property for the special graphs and . We note that each graph in here is perfect by . Actually, Eq. (5) implies a special case of the result in  since in this reference the authors proved that the lexicographic product is perfect iff G and H are perfect.
(by Theorem 1).
and (by Theorem 2).
(by Theorem 3).
(by Theorem 4).
(by Theorem 5).
(by Theorem 6).
(by Theorem 7).
Dedicated to Professor Hari M Srivastava.
The first, third and fourth authors are partially supported by Research Project Offices of Selcuk and Uludag universities. The second author is supported by the Faculty research Fund, Sungkyunkwan University, 2012 and Sungkyunkwan University BK21 Project, BK21 Math Modelling HRD Div. Sungkyunkwan University, Suwon, Republic of Korea.
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